Breakdown of blackjack losses
Quote: rickrandWhat would be the breakdown of losses in blackjack?
I understand that statistically if I play 1000 hands using basic strategy that I would lose 525 hands (52.51%) on the average.
How would those 525 losses break down? Meaning how many single losses, 2 in row, 3 in row, 4 in row, etc.?
It seems to me that in order for it to add up to 525 it would start with 262 single losses, 131 @ two in row, 65 @ three in a row...and so on until it adds up to 525. It this correct? Just divide in half down the line?
Your math is not correct.
Without doing a simulation, since double down and splits are real hands to consider, say my Grandma who plays BJ and NEVER splits and doubles.
525.1 average losses (using your numbers)
249.633 would be the average total of runs or streaks
(from exact calculations and verified by simulations)
# of runs is calculated from:
47.44% are streaks of exactly 1
52.56% are streaks of 2 or more
Distribution table of 525 losing hands in 1000 total hands played
| total losses | run | average | at least 1 streak % |
|---|---|---|---|
| 118.69 | 1 | 118.69 | 100.0000000000000% |
| 124.5 | 2 | 62.25 | 100.0000000000000% |
| 97.98 | 3 | 32.66 | 100.0000000000000% |
| 68.52 | 4 | 17.13 | 99.9999999999999% |
| 44.95 | 5 | 8.99 | 99.9999999408424% |
| 28.26 | 6 | 4.71 | 99.9975730557355% |
| 17.29 | 7 | 2.47 | 99.5563838806810% |
| 10.4 | 8 | 1.3 | 93.8855058648438% |
| 6.12 | 9 | 0.68 | 76.5750350133059% |
| 3.6 | 10 | 0.36 | 53.0883904837889% |
| 2.09 | 11 | 0.19 | 32.6791976771361% |
| 1.2 | 12 | 0.1 | 18.7114639908005% |
| 0.65 | 13 | 0.05 | 10.2866051854336% |
| 0.42 | 14 | 0.03 | 5.5316615772848% |
| 0.213 | 15 | 0.0142 | 2.9399129443708% |
| 0.1184 | 16 | 0.0074 | 1.5528046492025% |
| 0.0663 | 17 | 0.0039 | 0.8174870084971% |
| 525.0677 | 249.6355 |
also see
Streaks in blackjack ... some actual data ... Thread HERE
Quote: rickrandThanks so much for this breakdown. This clarified my issue. Thanks again.
You are welcome.
From reading your post in your Blog,
let me show you what you did and a formula that you can work out runs (streaks) with.
It is not an exact formula, but comes very close. The exact formula is a recursive one with many steps.
While only being close, for n<100, it is still very accurate.
In your Blog "That doesn't seem to jive with the wizards math of using .5251 x the ? power."
.5251^n is correct for just n.
example:
to lose 4 hands in a row is .5251^4 for EXACTLY 4 hands. (7.60% rounded)
to lose 4 in a row in exactly 5 hands is a way different calculation. (11.21% rounded)
A handy and accurate streak calculator can be found here:
http://www.pulcinientertainment.com/info/Streak-Calculator-enter.html
The math behind streaks or runs is a challenging exercise since it is not as easy to do as one would expect.
Now to duplicate the table I showed above.
We need these values:
p= probability of a loss; .5251
q= 1-p; 0.4749
n= number of trials; 1000
K= total number of runs (streaks) 249.65 how did we find that without a simulation?
r= run length. This comes later
Let us solve for K first.
=p^2+(p*q*n)
Now for a run of exactly 1
=K*q
That was easy. There is a different formula to use I could show here for the value of any "r"=run, but I will not right now. See if you can find the formula yourself.
If not continue.
Now for a run of exactly 2
r1(that you just calculated) - (r1*q)
Now for a run of exactly 3
r2(that you just calculated) - (r2*q)
etc. I think you can see the pattern.
I do it this way so you can create the below table in a spreadsheet.
The results are the average number of runs where n=1000.
r1 118.5442341
r2 62.2475773
r3 32.68620284
r4 17.16352511
r5 9.012567037
r6 4.732498951
r7 2.485035199
r8 1.304891983
r9 0.68519878
r10 0.35979788
r11 0.188929867
r12 0.099207073
r13 0.052093634
Compare this table to the other earlier in this thread and you will see the results are very close.
If you are lost, just play with the numbers, it will work for you. As can be seen I first calculate the average runs (streaks) per n trials.
There are a few formulas out there that do it a different way for coin tosses, but they all fail when I use them.
I know this can be simplified, but if someone else wants to do this, feel free.
Every time I post a table I get emails asking "How did you do it".
This will answer those that asked.
I know Honey, it is my turn to walk the dog after his bath!
Like teddys seemed to say, does losing both ends of a split count as 2 losses or one?
Needless to say a Grandma who never doubles or splits would very much change the W/L %'s. In other words, your 52.51% would change depending on the rules of the game in question. So at least know your W/L/T percents for the game you are playing to begin with.
And don't forget all net losses on a round are not created equal. A loss of 0.5 units in a game with surrender 3 times in a row is only 1.5 units. But, if DAS and re-splitting to 4 hands allowed, losing 24 units in the same 3 rounds sucks alot more, does it not lol? Streak analysis gives equal weight to each (each is counted as a loss in the 52.51%) despite one being a very much more rare event.
I like all this streak analysis and appreciate all the input.
Although I've always cared about chances of losing x units in y rounds more, including rounds with net pushes.
How about streak anlaysis that takes into account that rounds with a net win of zero actually do occur? When I track rounds, I count pushes as a round and LLLPPPL is a streak of 3, not 4. Mostly, I just don't care, except, maybe, for curiosity's sake. I care more about how many units I am down after those 7 rounds than the streaks within the 7 rounds.
Maybe I'm asking what is the point of streak analysis anyway? One's avg win when one wins is greater than one's avg loss when one loses. Doesn't that count for something?
Quote: rickrandI'm still trying to work out a model of 10,000 hands in blackjack and I'm running into conflicting information. Can you tell me how to model this?
I have gone through the painstaking process of actually playing 10,000 hands on three different software programs and writing the results. They are all very different. What I'm trying to do is come up with the mathmatical "average" model to compare to what the simulation results are showing. Also, It seems there is a difference between cd installed programs and online programs. Can you comment on this?
Just about all the BJ simulators I have seen really just are interested in stats on counting and a few others.
try:
http://www.bjsim.com/default.aspx
you can have a print out of every hand after you run a sim or one of theirs.
But you would have to place the data in Excel or something similar to find the results you are looking for.
Quote: rickrandWhat I'm looking for is this:
1) Total wins and total losses (supposed to be 52.5% & 47.5% - online programs ran above 55% losses in 10,000 hands, including the wizards practice site). 2) The breakdown of those numbers into their streaks. For example in 10,000 hands how many times will I lose 5 in a row, etc. I'm also trying to figure out how the 47.5% of wins are distributed.
PS: I've looked at all the calculator and data information links and still can't figure out how to model this. It seems like there would have to be a simple simluator out there (using basic strategy) that would spit out the results of 10,000 hands.
Here are my results for streaks. Let us see how close they come to actual sim results. In a simulation of 10,000 hands you will have more wins and losses total than just 10,000.
Win
4750 wins
2494 runs
run avg#
1 1309.4563
2 621.9352
3 295.3897
4 140.2961
5 66.6340
6 31.6480
7 15.0313
8 7.1391
9 3.3908
10 1.6104
11 0.7649
12 0.3633
13 0.1725
14 0.0819
15 0.0389
5250 Lose hands
2494 runs
runs avg #
1 1184.7938
2 621.9476
3 326.4899
4 171.3900
5 89.9708
6 47.2299
7 24.7932
8 13.0151
9 6.8323
10 3.5866
11 1.8828
12 0.9884
13 0.5188
14 0.2724
15 0.1430
I posted this graph that I made in an earlier discussion of streaks. It shows the probability of getting 2, 3, 4, ... 10 losses in a row out of some number of plays.
Most of the curves are for a zero house edge (i.e. tossing a fair coin). I expanded the case of 6 losses in a row to include two different house edges.
If you are having trouble reading the logarithmic graph, it says that you have a 50% chance of losing 6 in a row for a coin toss at just under 90 plays. If you are playing the pass/line bet at craps (6 PL)on the graph which has a 1.41% house edge you will get to 50% chance of losing 6 in a row at just over 80 plays. If you are making a red/black bet in American roulette(6 RL)on the graph , you have a 50% chance of losing 6 in a row at just about 70 plays.
Blackjack has a house edge fairly close to the pass line bet in craps. However, you can think of the heavy line (coin tossing) which has no house edge as an upper limit.
The number 6 losses in a row was chosen, because people who play Martingale will very often double their bets up to 6 times in a row. The most common ratio of max bet to min bet at a casino table only permits you to double your bet up to 6 times.
------------------------------
The case of a streak of two losses in a row has the actual percentages highlighted by little balls on the graph. As this case is fairly extreme, it's only practical interest is to see your if you can do the calculations by hand. The first four results are 25%=2/8, 37.5%=3/8, and 50%=4/8, and 59.375%=19/32. If you can't see the graph, you can download the graph as a .pdf file here.
Quote: rickrandThank you once again. As you can tell I'm in the early stages of what I'm sure is a long journey to understand this. Curious question: why does it work out that the amount of runs for both wins and losses is 2494?
You are welcome.
The total number of runs for both in a session can only be 1 off or equal.
w=streak of wins
L=streak of losses
does not matter what length w or L is.
wLwLwLwLwLwLwLwLwLw
the above sequence has runs (streaks) w=10; L=9
wLwLwLwLwLwLwLwLwLwL
the above sequence has runs (streaks) w=10; L=10
Quote: rickrandFYI: the 10,000 hand simulations I've done using the cd installed software are very close to your numbers. The online software results are way off (to the casino's favor). Thoughts?
I do not know what is way off.
Are you talking about wins and losses. streaks or all?
The average number of losses per 10,000 trials is 5250 (using your percentages) with a standard deviation of ~50.
The average number of wins per 10,000 trials is 4750 with a standard deviation of ~50.
Quote: rickrandThanks for continuing to explain things for me.
The online softwares, after 10,000 hands,, totaled 56.3% losses/43.7% wins on one and 55.8% losses/44.2% wins on the other.
The CD softwares were 52.5%/47.5% and 51.2%/48.8%.
It can also depend on if the dealer hits soft 17s, and the number of pushes
With these rules:
dealer stand all 17
10,000 sessions of 10,0000 hands
about 7% of the time average
44.37% win
55.62% lose
The online software sounds high but your sample size is way too low to draw good conclusions.
Online software may also shuffle after every hand but I do not know the effect on win/loss ratios.
The Wizard may have spoken about that at his WoO site.
Win/Lose rates can relate to streaks and runs but the rules of the game and playing with basic strategy effects the house edge and those two, and luck, actually can determine if you win or lose in any session.
Q - "I play the negative system in black jack meaning I double every time I lose until I Win. I wanted to what the odds are of losing 4,5,6,7,8,9 hands in a row? How many hands should I expect to play till I lost 8 hands which is my stopping point? - Jay from New Haven, Connecticut"
A - The name for this system is the Martingale. Ignoring ties the probability of a new loss for a hand of blackjack is 52.51%. So the probability of losing 8 in a row is .5251>8 = 1 in 173.
You are reading his answer right and I do agree with his statement.Quote: rickrandThe Wizard of Odds answered the below question in a way that doesn't make sense to me, or the math you provided (which I think yours does make sense). He basically told this guy he would average an "8 in a row" loss about every 173 hands.
Am I not reading it right or do you disagree? He said you can arrive at your losing streaks by multiplying your loss percentage (52.5%) by the "power" number of the streak. In this case to the 8th power.. Thoughts?
Q - "I play the negative system in black jack meaning I double every time I lose until I Win. I wanted to what the odds are of losing 4,5,6,7,8,9 hands in a row? How many hands should I expect to play till I lost 8 hands which is my stopping point? - Jay from New Haven, Connecticut"
A - The name for this system is the Martingale. Ignoring ties the probability of a new loss for a hand of blackjack is 52.51%. So the probability of losing 8 in a row is .5251>8 = 1 in 173.
I have been reading and studying runs (streaks) and patterns, Penney Ante, pattern matching, non-transitive relation, Conway’s Algorithm, average waiting time etc.
I have some excellent pdfs I have found on the subjects. I am on vacation currently and can post my pdfs in a few days if you want to read on the subject. My folder is a mess and some pdfs I do not know where they came from.
Yes, My Thoughts:
The calculation is to lose 8 hands in a row:
.525^8 = 0.00577131 = 1/0.00577131 or 1 in 173.27
this is ONLY TRUE when n=8. For exactly 8 consecutive hands.
Example: hands 1-8, 2-9, 3-10 ... 100-107, 101-108 ... 1000-1007, 1001-1008 etc. any consecutive sequence of 8 hands
(edit: This points out what is called overlap. The first 10 hands all lost and since there were 3 groups of 8 in those first 19 hands, you would count 3 for the number of runs of 8.)
The 1 in 173 calculation is for 173 "groups of 8 hands". Computer simulations back up this statement.
A quick 100k sim shows an average of 172.74
more results below
p= .5251
n=8
grouped data
items: 100000
minimum value: 1.00
first quartile: 50.00
median: 120.00
third quartile: 239.00
maximum value: 2110.00
mean value: 172.74
midrange: 1055.50
range: 2109.00
interquartile range: 189.00
mean abs deviation: 126.67
sample variance (n): 29824.58
sample variance (n-1): 29824.87
sample std dev (n): 172.70
sample std dev (n-1): 172.70
The Wizard gave a correct answer but left out some details, as most do, because a total complete answer can get confusing.
I know he understands this.
The probability increases for 8 or more in a row losses as the number of trials increases.
A few stats:
44 hands: 10.1208676433335% (1 in 10) for losing at least 8 in a row at least 1 time.
109 hands: 25.1135386516652% (1 in 4) for losing at least 8 in a row at least 1 time.
253 hands: 50.0170648199266% (1 in 2) for losing at least 8 in a row at least 1 time.
500 hands: 75.0166530326321% (3 out of 4) for losing at least 8 in a row at least 1 time.
1074 hands: 95.0138764075711% (95 out of 100) for losing at least 8 in a row at least 1 time.
The average wait time for losing 8 in a row is 1 in 362.7 hands with a standard deviation of 356.176
Also remember you can have more than 1 winning and losing hand in BJ with splits and can also win twice your wager with double down hands.
A BJ simulation would be a great way to go, most simulators are not set up to give all win/loss/streak stats unless you code a program yourself. The numbers above are good approximations for the game of BJ.
The math behind runs (streaks) and patterns is quite interesting and challenging and at times not very intuitive.
Google "Penney Ante" and read a few pdfs and you will understand more if you are interested.
Losing streaks will be seen, on average, more times with longer lengths than winning streaks.
I have a feeling pacomartin will be showing a graph of the most popular casino games and the losing streaks a player can expect to see.
Quote: rickrandYou are on vacation. Please go enjoy. You have been very helpful to me. Since I'm typing just one last verification of this info.
In the tables you posted before this post you gave me a win and loss breakdown for a 10,000 hand trial.
Am I reading it correctly to say in 10,000 hands I would lose 5 in a row, on the average, 178 times? (line 4 which I take it includes the single loss is 171, plus line 9 which is another 7, =178). And I can extrapolate for 100,000 hands or 1 million hands, basically? Thanks again and enjoy your vacation.
You are welcome.
I did not add a cumulative column to those tables, but yes you are correct.
If you add up 5 to 15 (for the losses) you should get
5 or more losses
189.391 for 10k
that is an average of 52.8 hands (actual EV 50.64) my figures have been rounded.
Yes to your 2nd Q.
1894.424, 100k
18944.748, 1million
I have a demo webpage, sill under construction, that will run sims for BJ losses to see the average number of hands for a particular pattern, including streaks. It works best with a Google browser, it is a snail (very slow) with Firefox or Internet Explorer because of the long wait times for 6 and 8 in a row. It is JavaScript based and is just a worksheet for now. I think it is a version #1 of many.
But it does work.
I will put it up for the day then take it down. You can save it if you wish.
And no malware or viruses.
HERE
Enjoying Vegas Baby!
Quote: guido111
The calculation is to lose 8 hands in a row: .525^8 = 0.00577131 = 1/0.00577131 or 1 in 173.27
this is ONLY TRUE when n=8. For exactly 8 consecutive hands.
The probability increases for 8 or more in a row losses as the number of trials increases.
A few stats:
44 hands: 10.1208676433335% (1 in 10) for losing at least 8 in a row at least 1 time.
109 hands: 25.1135386516652% (1 in 4) for losing at least 8 in a row at least 1 time.
253 hands: 50.0170648199266% (1 in 2) for losing at least 8 in a row at least 1 time.
500 hands: 75.0166530326321% (3 out of 4) for losing at least 8 in a row at least 1 time.
1074 hands: 95.0138764075711% (95 out of 100) for losing at least 8 in a row at least 1 time.
The average wait time for losing 8 in a row is 1 in 362.7 hands with a standard deviation of 356.176
Losing streaks will be seen, on average, more times with longer lengths than winning streaks.
I have a feeling pacomartin will be showing a graph of the most popular casino games and the losing streaks a player can expect to see.
Rick,
Here is a graph for your use. I highlighted the data points Guido tabulated above for you.
The 50%/50% is basically a limiting case. There is no house edge. The smaller and smaller your house edge in your game, the closer you will get to the limiting case.
Graph of 8 losses in a row for both casino probability of winning 52.5% and 50%.
The second graph is the same calculations for less than 44 plays. Instead of expressing the results as percentage, I expressed them as 1 in x number of plays. As you observed, the graph will begin with 1:173 . For the limiting case of no house edge the probability of throwing 8 losses in a row out of 8 turns is 1:2^8 = 256.
Graph of 8 losses in a row, zoom beginning for both casino probability of winning 52.5% and 50%.
The documents are public, so you can download it and print it if that helps.
Quote: guido111A few stats:
44 hands: 10.1208676433335% (1 in 10) for losing at least 8 in a row at least 1 time.
109 hands: 25.1135386516652% (1 in 4) for losing at least 8 in a row at least 1 time.
253 hands: 50.0170648199266% (1 in 2) for losing at least 8 in a row at least 1 time.
500 hands: 75.0166530326321% (3 out of 4) for losing at least 8 in a row at least 1 time.
1074 hands: 95.0138764075711% (95 out of 100) for losing at least 8 in a row at least 1 time.
A chart of the above probabilities is below.
Shows Blackjack streaks (of losses) from 2 or more up to 16 or more per n trials.
The table below is the results of 10,000 session simulations per n hands. (It takes a few seconds to see that the table is a few tables rolled into 1. I may re-do the table layout later)
It shows multiple streaks per n hands. (I used it to create the above chart)
Examples:
44 hand row.~10% chance of losing 8 in a row, or more, at least 1 time
~1.5% chance of losing at least 11, or more, at least 1 time
500 row is for at least 1 run of 8 or more losses= ~75%
2(right below 500) is for at least 2 runs of 8 or more losses= ~41%
3(right below 2) is for at least 3 runs or more of 8 or more losses= ~16%
Shows Blackjack streaks (loss and wins) from 2 or more up to 16 or more per n trials.
