Bovada pick'em poker no longer fair? Or just variance from hell?

I could rant and rave but I'm going to just show the true, accurate numbers and results from my play at their casino and you can make your own judgements. These are all my results on the .25 cent and .50 cent pick'em poker machine at Bovada since mid December. Keep in mind, proper play (which I can do in my sleep) should pay back the player 99.45%.

Dates........................... hands played .......... points................. . $ won/lost
12/17 .......................... 4359 ...................... -1300............... .... -575
12/19............................ 478 ...................... +500.................... +125
12/24- 12/25 ................ 6316 ..................... -3100 ................... -1250
1/5 - 1/7 ..................... 6523 ..................... -3200 .................. . -1000
1/11 -1/15..................... 8164 ...................... -1100 ............... ....-375
1/17 -1/21 ................... 9921........................ +160 ................ . +130
1/28 -2/4 .................... 17862...................... -3880.............. ..... -1170
2/10 - 2/11 ................. 7965 ..................... -1705 ................. -500
2/16 -2/18 .................. 8367....................... -3200 .............. .. -800
2/24 -3/3 .................... 5549....................... -500 .................. -175
3/12 - 3/16 ................. 5310....................... -175 ............... ... .-118.75
3/18 - 3/19.................. 17882..................... -1245................... -461.25
3/24 ........................... 3825..................... -2385.................. -678.75
3/31............................ 3548...................... -680 .................. . -170
4/6............................. 11633..................... -2126 ................... - 536.50

Most were sessions with matching bonuses and rollovers, in all 15 sessions with 13 losing ones and 2 small winners.

117,702 hands with a loss of 23,941 points for a ridiculous 95.93% payback. Total money lost was -$7,555.25 gross losses with $2477.52 earned in bonuses for a net loss of -$5,077.73

Now, perhaps this is just a real bad run of variance, but 95.93% back after 117,702 hands? Lets see where we stand since they changed the machines to the 99.45 payouts: 658,123 hands/ 3,290,615 pts bet/ 3,242,817 pts returned = 98.55%. If I broke down the details even further, you would find that I am below expectation for royal flushes,straight flushes, 4 of a kinds, full houses, flushes, and straights. Pretty remarkable if you ask me. But I am no math whiz (not a slouch either) like some are here, so I'd love to have the opinions of those qualified to give one as to these results in relation to acceptable variance.

Is it possible to be .90% off of expected payback after 658,123 hands on a fair game? I don't know. Perhaps it is but it sure seems borderline to me.

To be fair, I must state that years ago before they changed the machines and they paid 99.95%, I played 874,551 hands and had a payback of 100.16% and with bonuses netted a profit of $19,471 and they paid out promptly and hassle free whenever I requested a payout. I was not a bonus only player either. I played often without one and gave them plenty of action on horses and sports so I was not a bonus only casino nit.

This continuing saga of getting killed at this game and previous success at the 99.95 machine can only lead me to 3 conclusions.

A. The machines are still fair, and I'm just experiencing freakishly bad results still in the parameters of a fair game with random deals.

B. The game was changed when it went to 99.45 so as to make all premium paying hands pay out less often than expected to increase their take even more (perhaps caused by big liabilities to themselves in this game with bonuses)

C. My account is an account that has been made to lose consistently in the casino due certain factors.


If anyone else puts in similar volume with this same game and happens to know their results in a very accurate way, I'd love to hear some input.

I am in no way telling anybody they should or shouldn't play here, and I'm not saying I know definitively what is causing such bad results. If you're someone experiencing similar losses way out of proportion after a large sample, perhaps there is something that needs to be looked at closer. And if you're doing well at this game, congratulations, I hope you continue to do well, and let this serve as a warning that if you have a bad run, don't chase thinking you'll have to get some of it back because you just might run real bad like me for a LONG time.

I made a similar post a while back and since it has continued, I made this thread. This will be my last thread ever about this here. I decided to quit playing at their casino because it just seems like I'm throwing money away, but for anyone else who plays there, good luck and I hope you do well, and I hope I'm just being paranoid, and I hope that they are dealing an honest game, but I can no longer afford to see if it would ever wield results closer to expectation.

First, I'd like the record to show that the standard deviation in this game is 3.987966 and the exact return is 0.994466.

Second, can you show the number of units won/lost? In other words, what would have been the results if you flat bet?

The Wizard, as always, asks relevant questions and I am sure would get the correct answer as to rigged/unrigged gaming if he had all the info. The problem for the person making the allegation that the game may be rigged is that they need meticulous records in order for the issue to be properly examined.

Most of my play will be confined to properly regulated casinos with less play at NA and online establishments.

Unregulated is unregulated. It leaves more doors open to questionable things.

many, many years ago when I went to Vegas yearly, I played pick-em at the Vegas Hilton (now Vegas Hotel).

I have never had a winning session w/that game. :(
nowhere close to 99.4% return.
lowest variance my butt!

incidentally, the time I decided to quit that game forever and move to the 9/6 JoB, I hit a Royal within 10min.

Quote: 100xOdds

... I played pick-em at the Vegas Hilton (now Vegas Hotel).

I thought it was called the Westgate.

I don't know what you're exactly asking for with units won or lost but I thought that information was given as obviously I play max credits every spin. Every spin is 5 units bet. So 658,123 hands is 3,290,615 units bet and I had 3,242,817 returned for a payback of 98.55 and a loss of 47,798 units.

Here's a further breakdown of how many of each type of paying hand i hit and following in parenthesis will be the expected amount.

Hands 658,123
RF 1 (2)
SF 13 (17)
4Kind 270 (279)
Full House 1487 (1551)
Flush 2069 (2099)
Straight 3199 (3334)

trips,2pr, pr measured in unit % 79.32% (79.31)

Now, none of these numbers are by themselves all that out of line, in fact, the straights are the farthest off expectation being that the chances of having less than 3200 straights is 0.94%, and even that's not anything that alarming or out of whack. What creeps me out is the consistency of every paying hand from straights on up is oddly almost pattern like below expectation. If I were to cut the 658,123 hands into approximate thirds, you would find each third eerily similar in the % below expectation.

Now, If I were to combine these numbers with the 874,551 hands I played before they reduced the straights from 55 to 50, it's perfectly normal

Hands 1,532,674
RF 2 (4.6)
SF 44 (40)
4 kind 632 (650)
Full House 3615 (3611)
Flush 4945 (4888)
straight 7660 (7765)

I can't explain it, but the game used to have a certain "feel" to it, and ever since the change it seems different. Whereas there used to be the occasional super hot streaks and winning sessions were common, it almost never happens anymore and winning sessions are scarce. Whereas you used to get point "explosions", now almost every 4 of a kind is followed by you losing that 600 points back within the next 30-45 minutes. There are cold stretches where you lose 1000 points in less than a 1000 spins all the time, this used to be a rare occurrence. Now granted, straights going from 55 to 50 is a big change, but it shouldn't have the effect like it seems to have.

Perhaps this is nothing more than variance, and it probably is, but I've always had a "6th sense" for this kind off stuff and according to my radar something is not quite right. But your math skills far exceed mine, and I respect your opinion. If you think it's perfectly normal considering all the information, then it probably is.

Quote: shakhtar

I don't know what you're exactly asking for with units won or lost but I thought that information was given as obviously I play max credits every spin. Every spin is 5 units bet. So 658,123 hands is 3,290,615 units bet and I had 3,242,817 returned for a payback of 98.55 and a loss of 47,798 units.

Let's look at 3/24 as an example:

Dates........................... hands played .......... points................. . $ won/lost
3/24 ........................... 3825..................... -2385.................. -678.75

So, you're saying you bet 3,825 hands and lost 2385/5 = 477 total bets?

If we can divide points to get hands played, then why did the last day have a point win of -2126. It doesn't divide evenly by 5.

Quote: Wizard

Let's look at 3/24 as an example:

Dates........................... hands played .......... points................. . $ won/lost
3/24 ........................... 3825..................... -2385.................. -678.75

So, you're saying you bet 3,825 hands and lost 2385/5 = 477 total bets?

If we can divide points to get hands played, then why did the last day have a point win of -2126. It doesn't divide evenly by 5.

I played 3825 hands, which was 5 points per hand. So 19,125 points bet, 16,740 points returned = -2385. This particular day I lost -325 points at .50 c machine (2.50 per hand) and -2065 points at .25 c machine (1.25 per hand) which gave us a monetary loss of - $678.75

last day had a point loss of 2126. obviously i had a straight flush which paid back 1199. Kind of symbolic that I would play $250 with a $250 match bonus with a 40x rollover, get a straight flush, and still lose the whole $500 before rollover was met.

I don't know where the confusion lies, as it's pretty easy but perhaps I need to clarify better.

Every hand played is 1 hand betting 5 points (max). If I played 10 hands, and had 1 flush, 1 pair of Q's, and 8 blanks, I would have 50 points played, 85 points returned = +35 points.

I can't do anything if bets of different size are mixed together. If you want my help, then you'll have to separate everything according to the amount bet.

Quote: Wizard

I can't do anything if bets of different size of mixed together. If you want my help, then you'll have to separate everything according to the amount bet.

There are no different sized bets. All bets are 5 points max bets. Whether If I'm playing a .25 cent machine or a .50 cent machine is irrelevant. All hands are 5 point bets. My points expectation is going to be the same If I play .25 cent machines, .50 cent machines, .05 cent machines, or $1 machines.

If I play 2000 hands (5 point max every hand) at a .25 cent machine, my expectation is 9945 points.
If I play 2000 hands (5 point max every hand) at a .50 cent machine, my expectation is 9945 points.

All the data I have is for every single hand being a 5 point max bet. The monetary value of the machine ends up determining how much money you win or lose, but it has no effect whatsoever on the analysis of the game probabilities. My payback percentage has been calculated by points won/points bet, the $ amounts had absolutely no effect on those numbers.

Quote: shakhtar

There are no different sized bets. All bets are 5 points max bets. Whether If I'm playing a .25 cent machine or a .50 cent machine is irrelevant. All hands are 5 point bets. My points expectation is going to be the same If I play .25 cent machines, .50 cent machines, .05 cent machines, or $1 machines.

If I play 2000 hands (5 point max every hand) at a .25 cent machine, my expectation is 9945 points.
If I play 2000 hands (5 point max every hand) at a .50 cent machine, my expectation is 9945 points.

All the data I have is for every single hand being a 5 point max bet. The monetary value of the machine ends up determining how much money you win or lose, but it has no effect whatsoever on the analysis of the game probabilities. My payback percentage has been calculated by points won/points bet, the $ amounts had absolutely no effect on those numbers.

And from the OP:

Quote:

Is it possible to be .90% off of expected payback after 658,123 hands on a fair game? I don't know. Perhaps it is but it sure seems borderline to me.

If you're asking what seems to me to be a good and worthwhile question about a sponsor on this site, and the pre-eminent mathematician on here tells you what he needs to answer your question, which he seems to be willing to do for free, why on earth are you arguing with him about what information he needs to do the analysis? Either you want your question answered or you just came on here to trash Bovada; one is legitimate, the other is spam. I just don't get it.

Quote: beachbumbabs

And from the OP:



If you're asking what seems to me to be a good and worthwhile question about a sponsor on this site, and the pre-eminent mathematician on here tells you what he needs to answer your question, which he seems to be willing to do for free, why on earth are you arguing with him about what information he needs to do the analysis? Either you want your question answered or you just came on here to trash Bovada; one is legitimate, the other is spam. I just don't get it.

He did give sufficient information. Just ignore the last $ column and uses points only

Dates........................... hands played .......... points................. . $ won/lost
12/17 .......................... 4359 ...................... -1300............... .... -575

He played 4359 hands at 5 pts each, thats 21795 pts thru, and at 99.45% he should have lost 119.8725. instead he lost 1300.

Quote: shakhtar

last day had a point loss of 2126. obviously i had a straight flush which paid back 1199. Kind of symbolic that I would play $250 with a $250 match bonus with a 40x rollover, get a straight flush, and still lose the whole $500 before rollover was met.

$250 with 40x play through = $10000

Pick-em is 99.4%
$10k x .006 = $60 expected loss

How do you calculate standard deviation? (1sd, 2sd, 3sd?)

Quote: 100xOdds

$250 with 40x play through = $10000

Pick-em is 99.4%
$10k x .006 = $60 expected loss

How do you calculate standard deviation? (1sd, 2sd, 3sd?)

Don't forget cashback and other bonuses.

Quote: andysif

He did give sufficient information. Just ignore the last $ column and uses points only

Dates........................... hands played .......... points................. . $ won/lost
12/17 .......................... 4359 ...................... -1300............... .... -575

He played 4359 hands at 5 pts each, thats 21795 pts thru, and at 99.45% he should have lost 119.8725. instead he lost 1300.

I've played at Bovada a lot and I never once bet points. Like at most casinos, you bet money.

However, just for the sake of argument, could it be said that the OP lost 1300/5 = 260 units on 12/17? In other words a return of -260/4359 = -5.96%?

There is also still the issue that if every bet was 5 points, why isn't the points won/lost on the last day evenly divisible by 5?

Quote:

Dates........................... hands played .......... points................. . $ won/lost
4/6............................. 11633..................... -2126 ................... - 536.50

Quote: Wizard

There is also still the issue that if every bet was 5 points, why isn't the points won/lost on the last day evenly divisible by 5?

He mentioned he hit a straight flush that day, which pays 1199 credits on a 5-credit bet.

Quote: Wizard

I've played at Bovada a lot and I never once bet points. Like at most casinos, you bet money.

However, just for the sake of argument, could it be said that the OP lost 1300/5 = 260 units on 12/17? In other words a return of -260/4359 = -5.96%?

There is also still the issue that if every bet was 5 points, why isn't the points won/lost on the last day evenly divisible by 5?

You always bet points. When you click on the link to pick'em poker, the virtual machine appears. It is the 5 cent per point machine. You then have the option to play the .05 cent machine, or change it to a .25 cent machine, a .50 cent machine, or a $1 machine. I believe there is also a $5 machine but I would never play that high so I never go to it. The machine denomination is prominently displayed in yellow on the right side of the machine.

The money you have in the casino is shown in a small display window in the lower left hand corner. Let's say you have $300. On the bottom right hand corner are three chip symbols, a $5 chip, a $25 chip, and a $100 chip. The LED read out above that shows 0. You them click a chip and points will appear above it. For example, if I'm playing a .25 cent machine, and I click the $100 chip, 400 points will now show in the LED. If I play the .50 cent machine and click the $100 chip, 200 points will appear on the LED. So in essence, you are always betting points. Now, there are the back of 4 cards showing. You must make a bet, the button on the right says BET MAX, you push that and that will reveal 4 cards, the 2 you must keep, and the 2 you get to choose one of to also keep and also draw two more. This creates your final 5 card hand. If you started with 400 points, you now have 395 points and if your hand does not have a pair of 9's or better, you stay at 395 points. If I had made a flush, the machine will ask me if I want to play double or nothing with a YES/NO, option, which I ALWAYS opt no, and then your point total will adjust in the LED. In this example, let's say I made a flush on the first hand I played, I now have 470 points. If I want to turn that 470 points into cash, I must either hit the cash out button, or I simple exit the game and the money will appear in my casino balance.

So, once again, every hand I play is the same 5 point denomination. If I play 10,000 hands at the .25 cent machine (MAX BET every hand obviously) and I lose -1200 points (-$300), then I play 10,000 hands at the .50 cent machine and win +800 points (+$400), and I want to calculate the payback % to me over that 20,000 hands, it will be 99.6%. The fact that I happened to win money in this example is irrelevant.

So if we're now on the same page, all I'm really asking is where on the scale of variance is it when after 658,123 hands (That's obviously 3,290,615 points bet at max bet every spin) I have been payed back 98.55% on a machine that should pay back 99.45%.

If i had played 50,000,000 hands and got back only 98.55%, there is no doubt the something is fishy in Denmark. And I'm reasonable enough that I would never ask if I had only played 50,000 hands. But I thought a -.90% after 658,123 hands was enough hands with enough of a spread vs. expectation to at least ask the reasonable question as to it's veracity.

If I had made 200 dice rolls on "random" dice roller, and rolled only 18 sevens, I can do a simple binomial to figure the chances of rolling 18 or less sevens on 200 rolls of random dice is 1.13%. Hardly cause for alarm. I can also figure that if I rolled 2000 dice rolls on a random dice roller and rolled only 180 sevens, that the chances of rolling 180 or less sevens are far worse than a million to one, so I can conclude with a lot of certainly that this dice roller isn't truly random after 2000 rolls, whereas I could not make that claim over 200 rolls. But dice rolling is simple, and video poker with many different payouts with different frequencies of occuring make it much more difficult to grasp if 650,000 hands is a large sample, or just a small sample.

And as a footnote from the 4/6 results you quoted, that point loss should have been -2146, not -2126. In that case every hand was on a .25 cent machine so -2146 x .25 cent = -$536.50. And as someone else pointed out, the reason -2146 is not divisible by 5 is that one of the paying hands was a 1199 straight flush.

This is about as clearly as I can try to explain this. If you're still not understanding what I'm asking after this, I think I'll throw in the towel on this question.

Quote: shakhtar

You always bet points. When you click on the link to pick'em poker, the virtual machine appears. It is the 5 cent per point machine. You then have the option to play the .05 cent machine, or change it to a .25 cent machine, a .50 cent machine, or a $1 machine. I believe there is also a $5 machine but I would never play that high so I never go to it. The machine denomination is prominently displayed in yellow on the right side of the machine.
.

We call them CREDITS, not that I didn't know what you meant.

Okay. Here are my results:

Hands played	117702.00
Net win	-4788.20
Expected value per hand	-0.0055340
Expected net win	-651.36
Standard deviation per hand	3.9879660
Total standard deviation	1368.18
Win due to "luck"	-4136.84
Z score	-3.0236050
p value	0.0012489

So, the probability of doing this badly, or worse is 0.125%, or 1 in 801. Could have easily been ordinary bad luck.

Quote: Wizard

Okay. Here are my results:

Hands played	117702.00
Net win	-4788.20
Expected value per hand	-0.0055340
Expected net win	-651.36
Standard deviation per hand	3.9879660
Total standard deviation	1368.18
Win due to "luck"	-4136.84
Z score	-3.0236050
p value	0.0012489

So, the probability of doing this badly, or worse is 0.125%, or 1 in 801. Could have easily been ordinary bad luck.

My question is, at what point would you be concerned? No just talking about BV, I mean in general.

Quote: Wizard

Okay. Here are my results:

Hands played	117702.00
Net win	-4788.20
Expected value per hand	-0.0055340
Expected net win	-651.36
Standard deviation per hand	3.9879660
Total standard deviation	1368.18
Win due to "luck"	-4136.84
Z score	-3.0236050
p value	0.0012489

So, the probability of doing this badly, or worse is 0.125%, or 1 in 801. Could have easily been ordinary bad luck.

Thank you for calculating that.

If you could do 1 more calculation for me, and do the same thing for my totals for 658,123 hands, net win -9559.6 with an expected net win of -3642.05. I'd appreciate it .

By the way, congratulations on your sale of WizardofOdds.com, sounds like it was a good deal.

Quote: Wizard

Okay. Here are my results:

Hands played	117702.00
Net win	-4788.20
Expected value per hand	-0.0055340
Expected net win	-651.36
Standard deviation per hand	3.9879660
Total standard deviation	1368.18
Win due to "luck"	-4136.84
Z score	-3.0236050
p value	0.0012489

So, the probability of doing this badly, or worse is 0.125%, or 1 in 801. Could have easily been ordinary bad luck.

Since I'm curious, and I'm currently taking a Business Stats class, how did you arrive at the standard deviations? I think I know how you usually get it (subtract the mean from each data point, square, divide by n-1, square root), but what numbers are we working with in this example? Would the EV of -0.0055340 be the mean?

Quote: shakhtar

By the way, congratulations on your sale of WizardofOdds.com, sounds like it was a good deal.

Probably from all that "BV gaffing" you're experiencing ;)

Quote: shakhtar

Thank you for calculating that.

If you could do 1 more calculation for me, and do the same thing for my totals for 658,123 hands, net win -9559.6 with an expected net win of -3642.05. I'd appreciate it .

By the way, congratulations on your sale of WizardofOdds.com, sounds like it was a good deal.

Thanks and you're welcome.

Z statistic = -1.8290971
p value = 0.0336925

Quote:

Since I'm curious, and I'm currently taking a Business Stats class, how did you arrive at the standard deviations? I think I know how you usually get it (subtract the mean from each data point, square, divide by n-1, square root), but what numbers are we working with in this example? Would the EV of -0.0055340 be the mean?

I think you know how to get the standard deviation of a single hand. The multiply that by the square root of the number of hands. You don't do n-1 because we know the standard deviation exactly. If were using an estimate, then we'd do the n-1. The reason has something to do with degrees of freedom. I don't recall the specifics.

Quote: Wizard

Thanks and you're welcome.

Z statistic = -1.8290971
p value = 0.0336925



I think you know how to get the standard deviation of a single hand. The multiply that by the square root of the number of hands. You don't do n-1 because we know the standard deviation exactly. If were using an estimate, then we'd do the n-1. The reason has something to do with degrees of freedom. I don't recall the specifics.

The top pays screw up Z-score statistics for short numbers of hands (like 100k hands in VP), when evaluating risk based on RTP alone. I actually think that the OP's results are much less out on the tail of the curve than the Z-score indicates here, due to the variance that results from not hitting the top payouts. For BJ (sd = 1.15) or Bac (sd = 0.94), 100k hands is plenty. For VP, it's not nearly enough to reliably use the Central Limit Theorem to estimate risk based on RTP alone.

Quote: teliot

The top pays screw up Z-score statistics for short numbers of hands (like 100k hands in VP), when evaluating risk based on RTP alone. I actually think that the OP's results are much less out on the tail of the curve than the Z-score indicates here, due to the variance that results from not hitting the top payouts. For BJ (sd = 1.15) or Bac (sd = 0.94), 100k hands is plenty. For VP, it's not nearly enough to reliably use the Central Limit Theorem to estimate risk based on RTP alone.

Good point. In this case he was playing Pick 'Em Poker, where the royals contribute only 0.34% to the return, compared to about 2% in conventional video poker. I thought the Gaussian curve was enough to make a decent approximation. Considering I didn't get paid a dime, I think I met the call of duty in this situation. If I was getting paid, I would have done a more robust analysis.

Quote: Wizard

Good point. .

Hi Mike, I think you actually made this point to me when we were doing loss rebate stuff on the Revel 100%. 8-)

Quote: Wizard

Okay. Here are my results:

Hands played	117702.00
Net win	-4788.20
Expected value per hand	-0.0055340
Expected net win	-651.36
Standard deviation per hand	3.9879660
Total standard deviation	1368.18
Win due to "luck"	-4136.84
Z score	-3.0236050
p value	0.0012489

So, the probability of doing this badly, or worse is 0.125%, or 1 in 801. Could have easily been ordinary bad luck.

A more detailed simulation that takes the exact payout distribution into account indicates a -3.51 SD result or 1 in 4762 overall probability. In this simulation the end result - a loss of 4788 units or more after 117,702 hands played at 1 unit bet size - occured 21 times out of 100,000 trials. I do think that -3.51 SD is a suspicious result given the weight of the large data size.

Here are the correct input parameters on how anyone can repeat the same simulation at Beating bonuses website (link to simulator: http://www.beatingbonuses.com/simulator_java.htm )

Quote: teliot

The top pays screw up Z-score statistics for short numbers of hands (like 100k hands in VP), when evaluating risk based on RTP alone. I actually think that the OP's results are much less out on the tail of the curve than the Z-score indicates here, due to the variance that results from not hitting the top payouts. For BJ (sd = 1.15) or Bac (sd = 0.94), 100k hands is plenty. For VP, it's not nearly enough to reliably use the Central Limit Theorem to estimate risk based on RTP alone.

It seems to be a common misunderstanding that if you apply normal distribution approximation (CLT) to estimate a chance of loss on video poker or high-payout games, that it would always overestimate the chance of loss due to those big wins. Quite often it is the opposite because normal distribution approximation also overestimates the left (losing) tail, ie. the losing tail is in reality more constrained than predicted by CLT.

Here is a post I made a couple of years ago when I analyzed some of Galewind software's slots and posted critique that normal distribution approximation shouldn't be used to characterize the possible range of returns after #spins played in the game's help-files. Even though the topic was slots, it also applies to video poker or any high-payout games:

http://www.casinomeister.com/forums/showthread.php?t=49985&page=10&p=488457#post488457



In the above image the red bars represent normal distribution approximation and blue bars actual simulation results - on a Galewind software slot after 2,000 spins played.

From the image you will see that normal distribution approximation overestimates the probability of a loss whenever the RTP is lower than 85%. Furthermore if you go lower than RTP 70% then it's practically impossible to have this low RTP, yet the normal distribution still shows some possible probability mass at this RTP. Applied to OP's Pickem poker case, his result is somewhere at the very end of left tail where wizard's normal distribution analysis overestimated the chance of his losing figures.

Therefore it's critical that these kinds of questions like the OP's losing figure be analyzed via actual simulation and never resorting to normal distribution, because it's possible for the Z-score to show something like -3.5 SD result on a data which is much more extreme than that.