Defeating the $25 Table Minimum

There is a better strategy though: Never play at cold tables (-EV). Medium tables are break-even and hot tables are +EV

Mr. O: I reviewed your very thorough, hard work and can find NO fault in the statistical analysis. Is the resultant figure of -.25 actually 1/4 of 1% or is it a much higher number like -25%? The "wild" card in the overall scenario is with some good luck and a skewed table, is the hard 8. For me, a $27 loss on the easy 8 is tolerable and why we call it gambling. If one were to compare this play against a regular hand where the PL is $10 + odds of $15 + play on one other number $12 on the 6, a 7 out prior to hitting anything (fairly likely) results in a total loss of $37. Conversion of the point is at .8333 with 5 ways to win and 6 ways to lose. In terms of the real play I have witnessed, most players are playing PL + odds + at least 2 other numbers and sometimes a junk bet or two which obviously increases the amount at risk and can result in larger losses. My sense of things is that the majority of patrons of this forum who consider themselves in some kind of "expert" class play that way regardless of the savant math they espouse on HA/HE/variance/EV. What say you?

tuttigym

Mr Ace2: I agree with that great advice. So when you walk up to the table, how can one tell? To my knowledge, "hot tables" happen about 8% of the time as opposed to "choppy" tables about 60+%, and "cold" tables 30+%. Again, how does one determine what is going to happen in the future at that table?

tuttigym.

Quote: Ace2

All gambling in a casino is obviously -EV. My view is that the EV is so low (under half a percent) on properly played games such as Craps and BJ that it's actually one of the cheaper forms of entertainment available. Assuming you like gambling of course. It's so low that you could even beat the edge over a lifetime of playing via variance

Yep.

The house edge is very low on craps and blackjack. Obviously a person can crank it up with prop bets and other types of hedges or bets….but 1.41% (pass) and 0% (odds) is pretty good. Why screw with it?

If it’s high limits that is causing the problem…I’d simply pass from playing. I’m not that desperate to play where I would need to invoke some convoluted plan to keep me playing longer.

Nobody is getting rich gambling. It’s entertainment. I can play $1 bubble craps all day everyday in Vegas and keep it close to net zero. With some variance…I might get a month of being net positive.

It’s all about what you’re looking for. The guy playing 3 times a week likely has a different agenda than the guy playing once every 2 months on vacation.

Mr.O: I truly appreciate your explanation and patience. For me and the casino, the PL bet requires and commits the player to right side betting including free odds + additional numbers other than the point. Depending on the point that -EV gets larger in two ways: (1) 6 & 8 .833; 5 & 9 .333; 4 & 10 .50 and (2) each additional wager within a given hand increases any kind of breakeven possibility let alone an overall win for the hand. So as you have shown above the more -EV bets the greater the overall risk of large losses. As you are aware, the average hand in craps is about 5 rolls to conclusion of a 7 out or a point conversion. Therefore, a "choppy" table will and does kill exclusive right side betting. One other thing and this is not a criticism, the assumptions people make about any form of wager at the craps table is that the player is going to play and wager exactly the same for each hand so that the math and statistics will provide a "certainty" of outcome. That is not how I play, however, it is how most players play that I have witnessed.

For me, the doey/don't, regardless of the table minimum, gives me the freedom to chose based on how I feel the table is playing. After all, craps, like all games of chance, is a guessing game, and no amount of savant math will change that.

tuttigym

Quote: tuttigym

Mr Ace2: I agree with that great advice. So when you walk up to the table, how can one tell? To my knowledge, "hot tables" happen about 8% of the time as opposed to "choppy" tables about 60+%, and "cold" tables 30+%. Again, how does one determine what is going to happen in the future at that table?

tuttigym.

Enter a table when a shooter has won three points...that's a hot table. He's proven his ability and will almost certainly win another three. But get out after that because then a losing streak is overdue.

Quote: tuttigym

Mr. O: I reviewed your very thorough, hard work and can find NO fault in the statistical analysis. Is the resultant figure of -.25 actually 1/4 of 1% or is it a much higher number like -25%?

It is actually 25 cents, I'm surprised it's that small myself, and had been waiting for someone to point out a mistake. Bear in mind that before you add the hard 8 bet the expectation is a loss of zero, since we are looking at this at a certain stage of a doey-don't game that excludes the come-out.

Quote:

... In terms of the real play I have witnessed, most players are playing PL + odds + at least 2 other numbers and sometimes a junk bet or two which obviously increases the amount at risk and can result in larger losses. My sense of things is that the majority of patrons of this forum who consider themselves in some kind of "expert" class play that way regardless of the savant math they espouse on HA/HE/variance/EV. What say you?

I too have noticed a disregard for the best bets from those very same, we just heard about hedging with the 'any craps' on the come-out from someone who is handier with the math than I am. I also notice that if someone learned to play from other players and the dealers, they learn to love certain bets and can't abandon them later after accepting all the math.

Quote:

For me, the doey/don't, regardless of the table minimum, gives me the freedom to chose based on how I feel the table is playing. After all, craps, like all games of chance, is a guessing game, and no amount of savant math will change that.

At 80 yrs old I guess you can do as you please. As for your many other comments I'm not sure I have much to add at this point

Quote: tuttigym

Mission 146: Thanks for the post. What are AP's and who are "most of us"? I am not being critical. It is just that folks who consider themselves to be craps "officiandos", i.e. experts that I have seen play do not do much better than their novice counterparts even though they claim to know the best bets with the lowest HA/HE. It is gambling, and play along with the dice tosses are random events.

The "expected loss of their action" has mostly to with ego, lack of discipline, and the simple arithmetic associated with any gambling endeavor.

tuttigym

APs = Advantage Players

When it comes to advantage players and math guys, just because we discuss the EV doesn't generally mean that we have any ill-feelings against players that do not play in the way that yields the lowest expected loss. Absent something that would make a game positive (such as a promotion), the only way to play Craps with the lowest expected loss would be one of two ways:

1.) Find someone who trust who doesn't like making odds bets and get to bet their odds for them.

OR:

2.) Not playing Craps.

364 days per year, I choose Option 2...the other day of the year I will buy into Craps for something like $40, make a Pass Line bet of $5, Crap Check on the CO of $1, double odds, and Place the Six and Eight (unless one is the point) for $6 each. I'll also put the dealers on Six and Eight for $1 each because this particular casino pays them out at $3 and down when it hits, so I'll put them back up.

My expected loss on Pass Line is 7.05 cents, 15.2 cents expected loss on the Place Bets (when both need to be made, or 7.6 cents per), the $2 in dealer tokes is gone (and any time I put the $1 back up, which is whenever it hits, so those are fixed loss) and then I'm losing 11.1 cents on each Crap Check, which seems absurd when compared to to the other numbers!

If I seven out relatively early, then my expected loss is under $1, not counting the tokes, which is acceptable to me for getting to roll the dice in the rare event that I feel like it. If I am winning, then my expected loss goes up, because I'm making more bets than if I'd had an early seven-out.

If I make three points in a row, then I keep going. If I fail to make three points in a row, then I quit immediately. If I make three points in a row, the whole thing basically resets and now I am willing to have two seven outs before quitting. After the first one, I have to make three points in a row to bring it back up to two, that failing, quit.

Anyway, that's what I advocate for---I don't really want much more than for people to understand the expected loss of each bet and make an informed decision as to whether or not it's worth it to them. I'll eat my expected 18+ cents expected loss on the Come Out roll with that Crap Check because I don't like having a net loss on the Come Out roll. I'd rather win some amount or move on to trying to make my point.

I'd probably eventually abandon the Crap Check if I played Craps a lot, but fortunately for me, I've only played once in three years. Over the last seven years, I've probably averaged 1-2 sessions per year...I've basically also played whenever I have felt like it; I only rarely get the, 'Urge,' to shoot the dice. I figure my total expected loss is ~$100 in all of that time.

Also fortunately for me, I have a really low risk tolerance and an even lower tolerance for losing.

TDVegas: Your last three paragraphs are right on. I do not believe I have ever seen "bubble-craps," so i will take your word for it. As I might have mentioned before Biloxi's gaming is very competitive with most minimums at $10 or less. The Palace for instance has $3 craps tables whereas the Beau's tables are $10. You would be in hog heaven there alone with Mr. Chump c. Vegas may have the glitz and glamour, but Biloxi has the value and real economical gambling fun.

Without getting into a pissing contest, your 1st paragraph is misleading in that the HE, after the point is established, jumps progressively. It is not about a 0% HE on the payout; it's about the win/loss potential of the point conversion.

tuttigym

Mr. Ace2: The last time I saw someone win three points in a row was, I cannot remember, but there was a time it happened that I did see, and his 4th come out was point; 7out. It happens in a random game where the dice have no memory. All I am trying to say is that nothing in gambling is for certain. Sometimes we have to trust our instincts and cross our fingers.

Mr. Mission 146: I like your attitude and style. I wish you well.

tuttigym

5 out of 6 shooters don't make 2 points on their single turn. Should I Martingale the Don't Odds?

Mr. Tanko: Thanks for the input. I have ordered the WC Pro and tried to download it on my PC, however, the PC blocked the download saying it would harm my system. Very frustrating. I have contacted WC and they have responded, but so far their "fix" did not work. I am sure they will continue to help. I am waiting for the disc now.

When I run my "plan(s)" I do not do the same thing over and over again because I do not play the same way with each hand other than the come out. As I have stated on other posts, I am willing to forego the one roll player advantage in favor of what I perceive as wagering flexibility on either the right side or "dark" side. Unlike the "establishment" players, I am not out to take chances on large or multi-number bets because I set $$$ winning or losing goals. When either goal is met, I quit. I also do not leave my wagers on the table to eventually be swept away by a 7 out. It doesn't always work out, but most times it does.

tuttigym

Mr. Chump c: While you make a valid point, I really do not advocated personally multiple presses or Martingale. Conservative wins on multiple hands works well for me because there are also conservative loses during any session which I can live with and most times overcome within the session. Good question!

tuttigym

Quote: tuttigym

TDVegas: Your last three paragraphs are right on. I do not believe I have ever seen "bubble-craps," so i will take your word for it. As I might have mentioned before Biloxi's gaming is very competitive with most minimums at $10 or less. The Palace for instance has $3 craps tables whereas the Beau's tables are $10. You would be in hog heaven there alone with Mr. Chump c. Vegas may have the glitz and glamour, but Biloxi has the value and real economical gambling fun.

Without getting into a pissing contest, your 1st paragraph is misleading in that the HE, after the point is established, jumps progressively. It is not about a 0% HE on the payout; it's about the win/loss potential of the point conversion.

tuttigym

If your play is going to be a complex, convoluted “up and down” with wagers, hedges and other ‘nuances”….bubble craps is the best bet. Fast, quick. On a table….you’ll get eye roll galore from dealers and players if your plan slows the table to a crawl.

I see bubble craps players tapping the screen 20-30 times between bets. Each tap is making, moving, taking down or hedging bets. Quite the scene. Still…the bubble rolls every 30 seconds and waits for no one.

Quote: ChumpChange

5 out of 6 shooters don't make 2 points on their single turn. Should I Martingale the Don't Odds?

Good point..I never thought of it that way. Approximately .4^2 = 16% of shooters will win two or more points

Caveat: that calculation does not account for the table being hot, cold or choppy.

I'll have a hot roll making 5+ PL wins, then the dice go around the table like an ice cube on the "point-7" Titanic. I should play right-side when I shoot and Don't side after I shoot.

Mr. TDVegas: I don't think I have ever seen a craps bubble machine and by the way you describe it, I don't think I would like it. I need to be able to get my chips to the dealers and interact with them to do my thing. Tapping a screen to place or take down bets with my terrible hand/eye coordination would disastrous. Sorry about repeating myself.

tuttigym

Mr Chump c: Sounds like a plan. Buy in for $200 on a $5 or $10 table and try it.

tuttigym

I've seen a gaggle of 90 year olds at the craps table with the hot shots.

Mr Chump c: Did they have walkers and oxygen tanks?? If so, stay clear, one of those tanks will surely explode with a really "hot" roll.

tuttigym

Quote: ChumpChange

5 out of 6 shooters don't make 2 points on their single turn. Should I Martingale the Don't Odds?

I've been destroyed Martingaling the Don't Odds. I've also quadripled my buy in.

There's no system that works all the time!

Thank you all for your responses and comments. It is now time for me to produce my "plan," "strategy," "vision," and/or "scheme." Rather than make a huge lengthy post, which may be boring to some, I will submit my ideas in digestible pieces which can be easily discussed.

Caveats: (1) As previously stated, I have NOT tried or tested these wagering patterns at any casino or real craps table. (2) Betting scenarios described herein are rooted in fact and have been witnessed by most anyone who has played the game. (3) I make NO guarantees or warranties that any plan or system delivered here will result in riches beyond one's wildest imagination so that retirement to one's bucket-list destination will be stress and debt free.

Defeating the $25 table minimum form me starts with the Doey/Don't at $25 each plus $1 on the 12. After the point is established, the "establishment" player will place at least $25 free odds and usually two or three additional Place bets. The amount at "risk" and in play on the "right" side totals $125 or more. An immediate 7 out, which does happen, will result in a devastating loss of $125. Really tough to come back from that one. I, on the other hand, will cover those same four numbers (the point + three place bets) with a total wager of $6.00. How? By hopping the point, say the 5, ($2); the easy 4, 6, and 10 ($4). Now an immediate 7 out results in a $6 loss total plus perhaps $1 at Come Out on the 12. That circumstance is easily handled without indigestion or stress.

If the point is converted right away, the other player wins $50+ (PL + Odds) ; I win a net $11. My bets are down and the player still has three Place bets on the table at risk for the next hand. He must then replace his PL + Odds bet for the next hand and risk another $125. My question for any of you reading this is: How much has the player won or lost starting the third hand with the "new" bets on the table after collecting the $50+ from the previous hand?

I have much more to exhibit with additional hands to be played. I await your comments, slings, and arrows.

tuttigym

Quote: tuttigym

Thank you all for your responses and comments. It is now time for me to produce my "plan," "strategy," "vision," and/or "scheme." Rather than make a huge lengthy post, which may be boring to some, I will submit my ideas in digestible pieces which can be easily discussed.

Caveats: (1) As previously stated, I have NOT tried or tested these wagering patterns at any casino or real craps table. (2) Betting scenarios described herein are rooted in fact and have been witnessed by most anyone who has played the game. (3) I make NO guarantees or warranties that any plan or system delivered here will result in riches beyond one's wildest imagination so that retirement to one's bucket-list destination will be stress and debt free.

Defeating the $25 table minimum form me starts with the Doey/Don't at $25 each plus $1 on the 12. After the point is established, the "establishment" player will place at least $25 free odds and usually two or three additional Place bets. The amount at "risk" and in play on the "right" side totals $125 or more. An immediate 7 out, which does happen, will result in a devastating loss of $125. Really tough to come back from that one. I, on the other hand, will cover those same four numbers (the point + three place bets) with a total wager of $6.00. How? By hopping the point, say the 5, ($2); the easy 4, 6, and 10 ($4). Now an immediate 7 out results in a $6 loss total plus perhaps $1 at Come Out on the 12. That circumstance is easily handled without indigestion or stress.

If the point is converted right away, the other player wins $50+ (PL + Odds) ; I win a net $11. My bets are down and the player still has three Place bets on the table at risk for the next hand. He must then replace his PL + Odds bet for the next hand and risk another $125. My question for any of you reading this is: How much has the player won or lost starting the third hand with the "new" bets on the table after collecting the $50+ from the previous hand?

I have much more to exhibit with additional hands to be played. I await your comments, slings, and arrows.

tuttigym

I do not intend to whip out my slings and arrows over this, but merely to ask a few questions:

1.) Does the casino where you are playing (assuming the Hop & Hard minimums are even $1) have some rule that you must have a Line bet in order to make these bets? I ask because I have not played Craps at a great many locations, and only one that I fully understand the House Rules, which are that you're only required to have a Line Bet working if you want to SHOOT the dice---and only when it would be your turn to shoot.

If that is the case at the casino that you would visit, then you would never have to do your Doey-Don't + 12 Hedge to accomplish what you are suggesting.

2.) Why are you hopping the, "Easy," ways on non-point numbers? I know you are not big on expected loss, but making $25 Place bets on BOTH the six and eight has a lower expected loss, per roll, than the $4 you are exposing hopping those other easy ways that you mentioned.

Mission146: Really good questions:

(1) All the casinos I have played will not allow one to just play hop bets regardless of the table minimum. I tried that hopping the 7 with no line or PL bet even when I was NOT holding the dice. The table was really "cold," and I had won three consecutive times after the point was established and the pit boss stopped me on the fourth attempt to hop the 7 and told me that I had to have a table minimum bet on the line or a PL bet. The next hand I did the Doey/Don't and he really gave me the "look." I shrugged my shoulders; hopped the 7, and won again. I was following the rules, so he let it go. I did not overstay my welcome though and left the table shortly thereafter.

(2) Two reasons: (a) Hard numbers are a 35-1 shot; a really bad bet and (b) When one wins on a hopped number those extra $1 bets on the Hard number dilute the win. As for your "expected loss" math, if I am playing the same numbers as the player placing $25 and those numbers hit, then the so-called expected loss" is the same and my return of 15-1 is far greater than the 1+-1 on the regular place bet plus my risk exposure is 25x less to a 7 out.

I love the way you are thinking and challenging me.

tuttigym

Quote: tuttigym

Mission146: Really good questions:

(1) All the casinos I have played will not allow one to just play hop bets regardless of the table minimum. I tried that hopping the 7 with no line or PL bet even when I was NOT holding the dice. The table was really "cold," and I had won three consecutive times after the point was established and the pit boss stopped me on the fourth attempt to hop the 7 and told me that I had to have a table minimum bet on the line or a PL bet. The next hand I did the Doey/Don't and he really gave me the "look." I shrugged my shoulders; hopped the 7, and won again. I was following the rules, so he let it go. I did not overstay my welcome though and left the table shortly thereafter.

(2) Two reasons: (a) Hard numbers are a 35-1 shot; a really bad bet and (b) When one wins on a hopped number those extra $1 bets on the Hard number dilute the win. As for your "expected loss" math, if I am playing the same numbers as the player placing $25 and those numbers hit, then the so-called expected loss" is the same and my return of 15-1 is far greater than the 1+-1 on the regular place bet plus my risk exposure is 25x less to a 7 out.

I love the way you are thinking and challenging me.

tuttigym

1.) That sucks.

2.) I agree when it comes to the risk-exposure, but as you undoubtedly know, a long run of non-point, non-easy, non-seven rolls could eat you alive in a serious hurry on those easy hops.

Paying 15 to 1, each easy hop bet has an expected loss of 11.11 cents, which reflects the House Edge of 11.11%. You had mentioned hopping both possible fives, both easy ways to the six, as well as easy four and easy ten, which each have two ways of happening. The expected loss of that $6 exposed is 66.66 cents.

In the meantime, and I guess it would be $30 Place Bets to win $35, you have an expected loss of 0.46%, or 27.6 cents, on both bets combined. I'm typically not a huge fan of, "Per roll," (who really ever picks these back up before resolution) but consider it appropriate in comparison to hop bets.

Further, only the seven causes you to lose Place bets. The six easy hops have a combined twelve ways of occurring, so that means that 2/3rds of your rolls, or 24/36 combinations are losing. (1-1, 2-1, 1-2, 2-2, 3-3, 2-5, 5-2, 6-1, 1-6, 4-3, 3-4, 4-4, 5-3, 3-5, 2-6, 6-2, 6-3, 3-6, 4-5, 5-4, 5-5, 6-5, 5-6, 6-6).

With that, let's compare the probability of losing both Place Bets to the probability of losing the equivalent in your Hop Bets---which would be two Place Bets.

With both Place Bets, then you have 6/16 to, 'Immediately,' lose both...which just means a seven before either of the two hit. That's 37.5% of the time, so not great. Let's look at Place Bets hit and down to look at how these might work out:

Win Both Place Bets, Both Down: (10/16 * 5/11) * 70 = 19.8863636364 (Probability 0.28409090909)

Win One Place Bet, Seven, Both Down: (10/16 * 6/11) * (5) = 1.70454545455 (Probability 0.3409090909)

Lose Both Place Bets: (6/16) * (-60) = -22.5 (Probability 0.375)

Probability Sum: 0.375 + 0.3409090909 + 0.28409090909 = 1

Expected Return Sum: 19.8863636364 + 1.70454545455 - 22.5 = -0.90909090905

Relative to Bet: -0.90909090905/60 = .015151516 (This is the House Edge per bet Resolved and we are assuming both resolve)

Things get a little more tricky when we look at the Hop Bets because we win on one to lose on five for an overall profit of $10:

(12/36) * 10 = 3.33333333333

Alternatively, we can lose on all of them:

(24/36) * (-6) = -4

-4-3.33333333 = -.66666666

.6666666/6 = .11111111 (Which reflects the House Edge of 11.11%)

In order to exceed the expected loss of the Place Bets played to resolution, we would only have to lose twice on these Hop Bets. With that, we can look at the probability of different results based on two rolls, or winning a Hop Bet on the first roll which makes the second roll not necessary for these purposes.

(12/36) * 10 = 3.33333333

(24/36 * 12/36) * 4 = 0.88888888888 (You have lost six dollars followed by winning ten dollars for a net profit of four dollars)

(24/36 * 24/36) * (-12) = -5.33333333333

-5.33333333333+3.333333333+.888888888 = -1.11111111233 (This is our new expected loss, and is not double .6666666 because we do not necessarily bet $12 in this scenario, our average total bet is:

(12/36 * 6) + (24/36 * 12/36 * 12) + (24/36 * 24/36 * 12) = 10

-1.11111111233/10 = -0.11111111123 (Reflects the House Edge of 11.11%, errors due to rounding)

So, given the mere possibility of having to roll twice after a losing result (and having both lose), we see that our expected loss on the $6 bets has already exceeded that of playing BOTH $30 Place Bets to resolution.

It gets worse for the third trial and beyond because now you can no longer profit, assuming flat betting. You have already lost $12 and will only make $10 (on that roll) if the third roll is successful, for a net loss of $2.

In the meantime, we don't even expect each Place Bet (Taken individually) to resolve for slightly more than three rolls. More to the two roll point, the probability of EITHER Place Bet resolving in one roll, or both in the event of a seven, is 16/36, which means that about 31% of the time neither Place Bet (no seven, no six, no eight (20/36*20/36)) will have resolved after two rolls.

Furthermore, if you can make Place Bets without having a line bet out, then it will also eliminate your expected losses on PL, DP & $1 bet on Midnight, which I didn't even factor into this comparison.

Your, "Risk Exposure," is also not 25x less because you are making six Hop Bets for a total of $6, which still ignores Midnight. Assuming that $30 Place Bets on 6 & 8 would be made, that's a total of $60 in bets, so relative to the first roll you are betting 10% of what the Place Bets would be just in your $6 of Hop Bets. Of course, as you lose you will be making more bets, I assume, not that it matters if you do because you will be making another round of bets eventually, so it's all the same thing.

I don't know. I'd strongly prefer to do the $30 (each) Place Six and Place Eight and down on a win on either number. I understand the actual $$$ exposure at one time involved, but there comes a point that you're either there to gamble or you're not. I just feel like your strategy will lead to a ton of losing.

Or, better still, find a table with a lower minimum if it is possible. It would be great for all players who want a reasonable game if $25 tables anywhere but High Limit rooms would just be ignored by all.

I told you Mission was more adept with this kind of math

Quote: odiousgambit

I told you Mission was more adept with this kind of math

Aw, shucks!

Mission 146: There is no way that I can follow most of the above run of numbers. I realize that the hops are one roll bets and there could be a string without a hit, however, while you might be an anomaly regarding the one hit on the place bet and down, the vast majority of players leave their bets on the table until they are closed down with a 7 out. So do you take your $30 place bet and leave never to place it again? If you are playing the game, you will put it out there and be exposed. The major fallacy with math associated with HA/HE is not the payout, it is the actual edge the house has after a point is established. The oft stated 0% HA on the free odds bet is bogus because winning that bet must accompany a point conversion which is never in the player's favor.

Continuously citing "probabilities" Is truly meaningless because the reality of craps is the overall probability that after a point is established, if one is counting on point conversions, losing during any given session will happen well over 85% of the time.

Are you aware that one can hop the 7, after the point is established, 4 times for $3 each time and on the fifth roll, still win? I can place $6 in hop bets 21 times and lose all to equal one 7 out on the above $125 loss. Can that happen? Sure. But I won't go broke at the table on those 21 rolls, however, it is far more likely that there will be four 7 outs during that time generating much greater losses and possibly bankrupting the "establishment" player. On the other hand if that player, during that time, hits any place bets, and I am on those same numbers, I would win also. If I get two wins, I then would put $30 on the don't odds. Further wins would convert to regular $25 Place bets or I might wait for the next hand. My play is not static nor is it the same with every shooter.

Like you, I do believe in bringing down my bets after a limited number of rolls and wins. For example, placing bets across the board with a $4 Horn allows for me to hit two or three numbers and bring it all down. A 7 out is devastating and has happened, but that is gambling. But just to be clear, a $25 table limits my flexibility in the way I play and is not within my comfort zone. Also, one is not going to win anything of consequence during any session with $1 bets.

tuttigym

Quote: Mission146

-1.11111111233/10 = -0.11111111123 (Reflects the House Edge of 11.11%, errors due to rounding)

I couldn’t read through all this but it might be easier if you left numbers in fractional format. The above edge would be expressed as 1/9

Mr.O: His math is NOT real. The game and its actual outcomes during any given session ARE real.

tuttigym

Many casinos with a $25 minimum will not allow dollar bets on the hardways without at least one $25 bet on the layout.

Quote: tuttigym

Mission 146: There is no way that I can follow most of the above run of numbers. I realize that the hops are one roll bets and there could be a string without a hit, however, while you might be an anomaly regarding the one hit on the place bet and down, the vast majority of players leave their bets on the table until they are closed down with a 7 out. So do you take your $30 place bet and leave never to place it again? If you are playing the game, you will put it out there and be exposed. The major fallacy with math associated with HA/HE is not the payout, it is the actual edge the house has after a point is established. The oft stated 0% HA on the free odds bet is bogus because winning that bet must accompany a point conversion which is never in the player's favor.

Continuously citing "probabilities" Is truly meaningless because the reality of craps is the overall probability that after a point is established, if one is counting on point conversions, losing during any given session will happen well over 85% of the time.

Are you aware that one can hop the 7, after the point is established, 4 times for $3 each time and on the fifth roll, still win? I can place $6 in hop bets 21 times and lose all to equal one 7 out on the above $125 loss. Can that happen? Sure. But I won't go broke at the table on those 21 rolls, however, it is far more likely that there will be four 7 outs during that time generating much greater losses and possibly bankrupting the "establishment" player. On the other hand if that player, during that time, hits any place bets, and I am on those same numbers, I would win also. If I get two wins, I then would put $30 on the don't odds. Further wins would convert to regular $25 Place bets or I might wait for the next hand. My play is not static nor is it the same with every shooter.

Like you, I do believe in bringing down my bets after a limited number of rolls and wins. For example, placing bets across the board with a $4 Horn allows for me to hit two or three numbers and bring it all down. A 7 out is devastating and has happened, but that is gambling. But just to be clear, a $25 table limits my flexibility in the way I play and is not within my comfort zone. Also, one is not going to win anything of consequence during any session with $1 bets.

tuttigym

Quote: tuttigym

Mission 146: There is no way that I can follow most of the above run of numbers.

ALSO:

Quote:

Mr.O: His math is NOT real. The game and its actual outcomes during any given session ARE real.

tuttigym

Let's go ahead and address a few things here:

1.) So, you're writing the math off as, "Not real," even though, by your own admission, you cannot follow it? I think I'm going to start wholly dismissing everything that astrophysicists have to say about Universal event predictions, even though I don't understand about 75% of the way they get to their conclusions.

2.) There is no, "His math," it's just, "THE math." It applies to everyone and is not my property. It can be the math, "The way he did it," but it's immutable when it comes to expectation.

3.) The outcomes during a session reflect your actual results. Your actual results will not always be to expectation. If you make a single Pass Line bet, for instance, that bet will either win 100% or lose 100%, though it is expected to lose 1.41%...so a single Pass Line bet literally cannot perform exactly to expectation.

If everything went exactly according to expectation, then it wouldn't be gambling. That's what makes it gambling, otherwise, it would be a concept that you might have heard of called, "Subtraction."

Naturally, subtraction is what will ultimately happen sooner or later anyway.

Quote:

I realize that the hops are one roll bets and there could be a string without a hit, however, while you might be an anomaly regarding the one hit on the place bet and down, the vast majority of players leave their bets on the table until they are closed down with a 7 out. So do you take your $30 place bet and leave never to place it again? If you are playing the game, you will put it out there and be exposed.

I don't take down my Place Bets; I was suggesting that you COULD take down your Place Bets after a win. If you were to do that and (in the event that you win on both) can be patient enough to just wait until after the next Come Out roll, then I think you could sometimes enjoy long sessions on a relatively limited bankroll. I definitely think you'll give yourself a better chance to have a profitable session than this Hop idea.

As far as the rest of your post goes, please don't feel the need to defend yourself as I am not attacking. I don't care how you play as your money is none of my business. I simply wanted to demonstrate, for your potential benefit, the expected loss of your idea v. the Place Bets on Six & Eight.

Mr. Mission 146: First, if I sounded a bit chippy, it is out of my frustration with people using "THE math" as their total guidepost in explaining virtually anything in craps. I want to apologize. Second, "THE math" suggests, in the case of the PL bet, a 1.41% HA. Does that mean that over tens of thousands of PL bets that the player will only lose $1.41?? How is that possible on an even money wager on a $25; $15; or $10 table? It cannot happen. So while "THE math" is, by formula, correct, the reality of outcomes is NOT real. Besides, no one who plays the game just wagers exclusively throughout the totality of a session on just the PL. Because of that, why continuously tout the 1.41 HA? When will those who embrace such tell us all the HA on the PL bet AFTER a point is established? That might truly reflect craps HA reality because that is when the game actually starts.

The term "expectation" is vague and indefinite. It is supposition or guessing and truly unscientific. So to firmly associate that term with "THE math," by definition makes "THE math" unreal.

Because the hop bets are one roll wagers, it is most likely an overall loser vs the standard Place bet. I agree with that statement. The original exercise was to try to thwart the casino with a high table minimum and play with a smaller buy-in. So I demonstrated that.

So when you or Mr. O or Mr. Chmp c or anyone participating in this thread go to the casino, do you make your way to the craps table? Buy-in with how much? and play?
Are you an "establishment" player, i.e. PL + odds + Place bets or are you unorthodox in your gambling or does "THE math" have you so fearful that do not play at all?

tuttigym

Quote: tuttigym

"THE math" suggests, in the case of the PL bet, a 1.41% HA. Does that mean that over tens of thousands of PL bets that the player will only lose $1.41??

The math suggests that the player will lose approximately 1.41 cents per dollar wagered.

Mr.Dieter: So, again, how does one lose $1.41 on an even money bet on a $10 table and WHAT is the HA on PL bets AFTER THE POINT IS ESTABLISHED (emphasis)??

Better yet, tell us exactly how many PL bets one must make in order to lose that $1.41. If "THE math" is real, that number should be an absolute constant, i.e., X number of rolls in EVERY set EVERY time one engages in such a set. (Each roll of the dice is a come out roll only.)

tuttigym

Quote: tuttigym

Mr. Mission 146: First, if I sounded a bit chippy, it is out of my frustration with people using "THE math" as their total guidepost in explaining virtually anything in craps. I want to apologize. Second, "THE math" suggests, in the case of the PL bet, a 1.41% HA. Does that mean that over tens of thousands of PL bets that the player will only lose $1.41?? How is that possible on an even money wager on a $25; $15; or $10 table? It cannot happen. So while "THE math" is, by formula, correct, the reality of outcomes is NOT real. Besides, no one who plays the game just wagers exclusively throughout the totality of a session on just the PL. Because of that, why continuously tout the 1.41 HA? When will those who embrace such tell us all the HA on the PL bet AFTER a point is established? That might truly reflect craps HA reality because that is when the game actually starts.

The term "expectation" is vague and indefinite. It is supposition or guessing and truly unscientific. So to firmly associate that term with "THE math," by definition makes "THE math" unreal.

Because the hop bets are one roll wagers, it is most likely an overall loser vs the standard Place bet. I agree with that statement. The original exercise was to try to thwart the casino with a high table minimum and play with a smaller buy-in. So I demonstrated that.

So when you or Mr. O or Mr. Chmp c or anyone participating in this thread go to the casino, do you make your way to the craps table? Buy-in with how much? and play?
Are you an "establishment" player, i.e. PL + odds + Place bets or are you unorthodox in your gambling or does "THE math" have you so fearful that do not play at all?

tuttigym

Here's the thing about it when you want to argue against the math or that the math is not relevant:

1.) In an earlier post, I acknowledged that the expectation will not come to fruition in the context of a single bet. If you bet $100 on the Pass Line, assuming no odds, then you can only win or lose $100---you cannot lose the expected amount of $1.41.

Most people who disregard expectation think they are making some great point that renders the math meaningless when what they are saying is obvious. Anyone presenting the math would say that it's obvious. You can't lose $1.41 on a $100 Pass Line bet.

The more trials of a game that you have; the more likely that the sum of your results will be closer in line to the expected loss. In the case of a single $100 Pass Line bet, you're either running approximately $101.41 better than expected, after resolution, or approximately $98.59 worse than expected.

If the math is insufficient for you and you would like to do an Empirical observation on what is, "Real," then all you need do is go to a physical Craps Table and watch it for 10 hours or so, per day, for thirty days. Count the number of Pass Line bets that are made and index them as to all being for the same amount. Count the number of times the Pass Line wins and the number of times it loses. You will find that the results on those bets only (if you treat all players as having bet the same amount) will be really darn close to a 1.41% loss.

Or, you can acknowledge that the math works which, for most people, is much faster to do.

2.) I'm not, "Touting," anything about the Pass Line. As an alternative to your proposal, I liked the idea of making equal Place Bets on the 6 & 8 for you because I feel like it has a good chance of accomplishing what you seem to wish to get out of a session while giving you a reasonable probability of a profitable session.

3.) The term, "Expectation," is neither vague nor indefinite. (Amount to be Won * Probability of Winning) - (Amount To be Lost * Probability of Losing) = Expectation, at least, when it comes to a bet with a binary result, which the vast majority of Craps bets do have.

4.) If you want to thwart the casino, then I would recommend simply not playing at a $25 minimum. If nobody would play $25 minimums, then eventually, they would either have to get rid of Craps or reduce the minimums. It doesn't matter HOW you play a $25 minimum table, if you play it at all, then they are expected to win.

You're not, "Thwarting," anything by playing a game that you are expected to lose and by making high house edge bets. The only thing you're accomplishing is having a lower absolute loss on any given roll.

5.) I already detailed how I play Craps. My sub-optimal bets are my Place Bets and my terrible bet is that I have a, "Crap Check," for $1 working every Come Out roll. I have analyzed the expected loss on these bets and find the sum of my expected losses acceptable given:

A.) How rarely I play Craps.

AND:

B.) The most, in terms of actual money, that I am willing to lose when I do play Craps.

The math says what the math says. Nobody is demanding that you play in the mathematically optimal way. In fact, if you're playing at all, you're not playing in the mathematically optimal way. The mathematically optimal way to, 'Play,' a negative expectation game is NOT to play it at all.

Quote: tuttigym

Mr.Dieter: So, again, how does one lose $1.41 on an even money bet on a $10 table and WHAT is the HA on PL bets AFTER THE POINT IS ESTABLISHED (emphasis)??

Better yet, tell us exactly how many PL bets one must make in order to lose that $1.41. If "THE math" is real, that number should be an absolute constant, i.e., X number of rolls in EVERY set EVERY time one engages in such a set. (Each roll of the dice is a come out roll only.)

tuttigym

What is the House Advantage on Don't Pass bets AFTER THE POINT IS ESTABLISHED? There isn't one. A point having been established, Don't Pass is expected to profit, which is the precise reason that a casino will generally allow you to pick up a DP bet after a point is established, but not a PL bet.

To answer your question:

https://wizardofodds.com/games/craps/basics/#defining-the-house-edge

I would say that the HA on a PL bet AFTER A POINT IS ESTABLISHED does not change, because the HA is predicated upon the fact that you are more likely to win a PL bet on the CO roll than you are to lose.

There is a separate bet that can be made on the point after it has been established and that is called a, "Put Bet," the House Edges for which you can find on the link above.

Your second paragraph is nonsense due to a mathematical concept called variance. If there were, "Absolute constants," in terms of actual results, then it wouldn't be gambling...it would just be subtraction.

That's like saying a 50% probability of heads on a coin flip doesn't mathematically exist because a single flip is either going to be heads or tails. It would be like asking, "How many coin flips until the number of heads and tails flipped are going to be equal as an, 'Absolute Constant,'" there is no such number. That's why you get into concepts, given a particular number of bets made, such as standard deviation.

Giving you the base expectation is as simple as gambling mathematical concepts get. We can get into higher level mathematical concepts, but if you cannot accept expectation being what it is, then you'll consider the higher level concepts similarly irrelevant, or alternatively, will not understand them as you stated (by your own admission) that you could not follow my math vis-a-vis expectation.

I also can't emphasize enough that I don't care what you decide to do with your own money. If we were at the Craps Table and you said, "Give me $110 inside, a nickel on every hard way, two dollar midnight, two dollar snake eyes, and a $5 Yo," whilst tossing in $139 in chips, I wouldn't say a word to you as it's none of my concern what happens with your bets. If nothing else, I'd hope to go on a serious heater and then maybe you'd toss a shooter toke my way!

But, this isn't a Craps Table. This is a gambling forum that is a spin off of the WizardofOdds site which has always been primarily focused on mathematical expectation and identifying the lowest house edge bets. Some of us talk about the underlying math here. That's what I am here to do. If you don't want to play in a fashion that has the lowest expected loss, then that's your choice.

Honestly, if you understand the expectation when it comes to the bets that you are proposing and decide you want to do that---I couldn't be happier for you. I'm happy that you understand the expectation of the way that you are choosing to play and can sleep easily knowing I have done my job.

Quote: tuttigym

Mr.Dieter: So, again, how does one lose $1.41 on an even money bet on a $10 table and WHAT is the HA on PL bets AFTER THE POINT IS ESTABLISHED (emphasis)??

Better yet, tell us exactly how many PL bets one must make in order to lose that $1.41. If "THE math" is real, that number should be an absolute constant, i.e., X number of rolls in EVERY set EVERY time one engages in such a set. (Each roll of the dice is a come out roll only.)

tuttigym

It means if you made 10,000 of them the most likely outcome is you have lost $1,410 on your handle of $100,000.

$1.41 in losses is tied to $100 in action on the pass line. That’s what you are expected to lose.

$14.10 on $1,000 in action.

$141 on $10,000 in action.

$1,410 on 100,000 in action.

Quote: tuttigym

Mr.Dieter: So, again, how does one lose $1.41 on an even money bet on a $10 table and WHAT is the HA on PL bets AFTER THE POINT IS ESTABLISHED (emphasis)??

Better yet, tell us exactly how many PL bets one must make in order to lose that $1.41. If "THE math" is real, that number should be an absolute constant, i.e., X number of rolls in EVERY set EVERY time one engages in such a set. (Each roll of the dice is a come out roll only.)

tuttigym

The rules of the game are structured that you lose a few more times than you win.

$1.41 isnt an even multiple of $10. You will not be exactly $1.41 down after resolving 10 $10 pass line bets, despite having wagered $100 against a house edge of 1.41% and 1.41% of $100 being $1.41.

I would expect you to be down about 14 bets after 1000 trials.

Quote: Mission146

1.) In an earlier post, I acknowledged that the expectation will not come to fruition in the context of a single bet. If you bet $100 on the Pass Line, assuming no odds, then you can only win or lose $100---you cannot lose the expected amount of $1.41

.

I agree to your having acknowledge the above. There is no disagreement here.

Quote: Mission146

Most people who disregard expectation think they are making some great point that renders the math meaningless when what they are saying is obvious. Anyone presenting the math would say that it's obvious. You can't lose $1.41 on a $100 Pass Line bet.

"Most people" reading about the 1.41 HA somehow really believe that they will ONLY lose 1.41% of their bets on the PL. That is not your fault. It has become THEIR expectation. And I know you or anyone else cannot control their mathematical acumen. It is what it is.

Quote: Mission146

If the math is insufficient for you and you would like to do an Empirical observation on what is, "Real," then all you need do is go to a physical Craps Table and watch it for 10 hours or so, per day, for thirty days. Count the number of Pass Line bets that are made and index them as to all being for the same amount. Count the number of times the Pass Line wins and the number of times it loses. You will find that the results on those bets only (if you treat all players as having bet the same amount) will be really darn close to a 1.41% loss.

You have been told that, but you do not know that, and of course, I wouldn't do that nor would you to "prove" that to be true. Even the Wizard acknowledged what you are saying 10+ years ago in a post to me that it would take 250,000 rolls to empirically prove that statement, but he doesn't or can't know for sure.

Quote: Mission146

Or, you can acknowledge that the math works which, for most people, is much faster to do.

Again, "most people" don't even know about the math, and truthfully, probably would not comprehend it or care about it.

Quote: Mission146

2.) I'm not, "Touting," anything about the Pass Line. As an alternative to your proposal, I liked the idea of making equal Place Bets on the 6 & 8 for you because I feel like it has a good chance of accomplishing what you seem to wish to get out of a session while giving you a reasonable probability of a profitable session.

3.) The term, "Expectation," is neither vague nor indefinite. (Amount to be Won * Probability of Winning) - (Amount To be Lost * Probability of Losing) = Expectation, at least, when it comes to a bet with a binary result, which the vast majority of Craps bets do have.

I agree with the above two statements.

I have already stated that I would NOT play a $25 table. A $15 table would be my personal limit. Walking away for me is easy.

I totally agree that gambling in any form is personal and should not be criticized or mocked.

tuttigym

Quote: Dieter

The rules of the game are structured that you lose a few more times than you win.

$1.41 isnt an even multiple of $10. You will not be exactly $1.41 down after resolving 10 $10 pass line bets, despite having wagered $100 against a house edge of 1.41% and 1.41% of $100 being $1.41.

I would expect you to be down about 14 bets after 1000 trials.

These are only expected results, based on the odds. With variance…anything can happen. Over prolonged play, years, decades….I’m going to suggest it’s going to be closer to $1.41 than not.

Quote: Mission146

What is the House Advantage on Don't Pass bets AFTER THE POINT IS ESTABLISHED? There isn't one. A point having been established, Don't Pass is expected to profit, which is the precise reason that a casino will generally allow you to pick up a DP bet after a point is established, but not a PL bet.

I did not ask about the HA on Don't Pass Bets

Quote: Mission146

I would say that the HA on a PL bet AFTER A POINT IS ESTABLISHED does not change, because the HA is predicated upon the fact that you are more likely to win a PL bet on the CO roll than you are to lose.

One cannot just shrug off the most VALID question of this discussion by saying nothing changes regarding the PL bet after the point is established. That bet is the 800 lb. gorilla in the room; the albatross around the player's neck, the absolute reason "THE math" surrounding the 1.41% HA is NOT reality. The House mandates that the PL bet must stay on the table knowing that winning for the player has flipped big time in favor of the house. "Does not change."? Everything changes and by dismissing the question and the real answers is a disservice. Why not acknowledge the, in any discussion of the PL HA, the real pitfalls of the bet so that the uninitiated might be fully informed, which is your stated mission?

Quote: Missioon146

Your second paragraph is nonsense due to a mathematical concept called variance. If there were, "Absolute constants," in terms of actual results, then it wouldn't be gambling...it would just be subtraction.

I always though true math was an "absolute constant" once proven.

Quote: Mission146

If nothing else, I'd hope to go on a serious heater and then maybe you'd toss a shooter toke my way!

That would be my pleasure.

Quote: Mission146

This is a gambling forum that is a spin off of the WizardofOdds site which has always been primarily focused on mathematical expectation and identifying the lowest house edge bets. Some of us talk about the underlying math here.

I agree, but that discussion needs to be completely transparent including the good, bad, and ugly.

Quote: Mission146

Honestly, if you understand the expectation when it comes to the bets that you are proposing and decide you want to do that---I couldn't be happier for you. I'm happy that you understand the expectation of the way that you are choosing to play and can sleep easily knowing I have done my job.

I respect your intellect and attitude.

tuttigym

Quote: unJon

It means if you made 10,000 of them the most likely outcome is you have lost $1,410 on your handle of $100,000.

Got it. Except the PL bet does not stand alone in the game. It continues on until the hand is resolved. That process changes everything.

tuttigym

Quote: tuttigym

Got it. Except the PL bet does not stand alone in the game. It continues on until the hand is resolved. That process changes everything.

tuttigym

That is all calculated into the 1.41%.

I was just thinking that if I bet $10 PL & $10 odds, the HA after 72 come-out rolls is $10. If I bet $20 PL & no odds, the HA after 72 come-out rolls is $20. But that makes a 1.389% HA so my calculation is a little off.

If you've set up 60 points, all evenly spaced over the 6 box numbers, you'd have 9 numbers on the 4 & 10 and they would have lost 6 times each. On the 5 & 9, you'd have 10 points on each and lost 6 times on each. On the 6 & 8, you'd have 11 points on each and lost 6 times on each. So you've lost 6 times on each number times 6 numbers or 36 times, and you've won 24 times. So you're running a 12 win deficit on put bets and PL bets after they are established. You can take odds that will pay an amount to make the line bet loss seem less significant to the variance. All that averages out to what a 5 or 9 would pay (1.5X).

If you bet 4X Odds on the 4, 10, you'd have $10 PL + $40 Odds pays a total of $90, which is what you'd win if you made a $50 bet on the PB 4 or 10 (not a Buy Bet!)
If you bet 4X Odds on the 5, 9, you'd have $10 PL + $40 Odds pays a total of $70 which is what you'd win if you put $50 on the PB 5 or 9.
If you bet 5X Odds on the 6, 8, you'd have $10 PL + $50 Odds pays a total of $70, which is what you'd win if you made a $60 bet on the PB 6 or 8.

On Bubble Craps, when you Buy the 5 or 9, it pays 1.45X.
If you bet 9X Odds on the 5, 9, you'd have $10 PL + $90 Odds pays a total of $145, which is what you'd win if you bought for $100 on the 5 or 9.

Quote: ChumpChange

I was just thinking that if I bet $10 PL & $10 odds, the HA after 72 come-out rolls is $10. If I bet $20 PL & no odds, the HA after 72 come-out rolls is $20. But that makes a 1.389% HA so my calculation is a little off.

Two questions in the reality of a session or consecutive sessions: 1: Would you or anyone ever play that way? and 2. Why?

Quote: ChumpChange

If you've set up 60 points, all evenly spaced over the 6 box numbers, you'd have 9 numbers on the 4 & 10 and they would have lost 6 times each. On the 5 & 9, you'd have 10 points on each and lost 6 times on each. On the 6 & 8, you'd have 11 points on each and lost 6 times on each. So you've lost 6 times on each number times 6 numbers or 36 times, and you've won 24 times. So you're running a 12 win deficit on put bets and PL bets after they are established. You can take odds that will pay an amount to make the line bet loss seem less significant to the variance. All that averages out to what a 5 or 9 would pay (1.5X).

Two questions: 1: Have you or anybody you are acquainted with play that way or EVER witnessed or attempted that strategy? If so, 2: What were the actual results, and if not, why propose a betting plan that has virtually no chance of succeeding? But, in your defense, being outrageous can be lots of fun.

tuttigym

Quote: TDVegas

$14.10 on $1,000 in action.

I have bought in for $1,000 and never lost just $14.10, but then, I don't play that way nor have I seen anybody who might play that way lose that amount. With that $1,000 buy-in, how long would you estimate it would take to lose $14.10, and how many rolls or hands would be required to "prove" your claim, exactly?

Based on your answers to the above, the $10,000 or $100,000 examples would probably be so shocking that the edges of believability would be stretched beyond comprehension.

Quote: TDVegas

$141 on $10,000 in action.

$1,410 on 100,000 in action.

I just played for 10 hours of Bubble craps on a $30 buy-in and won 22 cents. So I cashed out before I was down to under a dollar again, but hadn't quite doubled my money. I'm sure there were several hundred rolls. I was betting $2 on the PL, or to switch things up $1 PL + $1 Odds. Sometimes I got down to "case bets" on the 4 or 10 and bet double odds and won (or lost). If I bet $2 PL + $4 Odds, I'd win $10 and get my $6 back. Do that 3 times quickly I can raise my bet to $1 PL + $2 Odds, or go for a $3 PL.

I've got a WinCraps chart up and I was just betting PL with Odds. It says there were 188 wagers on the PL with 98 wins & 90 losses, for a net win of 8 or 4.25%. I bet a total of $44,700 and won $22,740 & lost $21,960 for a net win of $780 on the PL or 1.745%. On the PL with Odds section, it says there were 132 wagers, 60 wins, 72 losses for a net loss of 12 on the odds points, or -9.091%. It says I wagered $94,860, won $61,974 and lost $52,920, for a net win of $9,054, or 9.545%. I think I was betting triple odds in the triple digits (like 10X a $15 table). So I'm up $9,834 on a $10K buy-in; I was up $21K halfway through. This was over 662 rolls in a sim period of 5.5 hours. I was also trying a new strategy so these are the good results before Win Craps inevitably goes bonkers negative on me after it learns my strategy.