# Determining the statistical significance of the performance of a gambler

Imagine someone claims they win significantly more than they lose when betting on roulette. Presuming that it were possible to have a winning system how could you calculate the statistical significance of their claim?

The roulette table has no "edge" (i.e. no zero/double zero...).

The issue comes with the fact that they are able to change their stake in addition to their odds. Even if they were hopeless at guessing and were wrong far more often than not, provided they only placed small losing bets and very large winning bets they would still come out on top. Their largest bets are 10, their smallest is 1, they are not simply making ever greater bets to recoup their losses.

I do not believe there is such a thing as a winning strategy and have no intention of testing this, so please give your answer safe in the knowledge that it won't inspire me to gamble myself into the poor house.

• The only objective way would be to get an honest accounting of gambling losses and gains for consecutive weeks of gambling activity. Then do a test to see if the mean of that distribution is significantly greater than 0. // A relative of mine had a habit of bragging about gambling gains on trips to Las Vegas, but never mentioning gambling losses. Going just by his reports you might be led to believe he was very successful at gambling. Hence the phrase "honest accounting" above. – BruceET Jun 11 '18 at 22:57

Suppose John deludes himself, thinking he has a 'system' for playing roulette. He plays 100 sessions, taking his lucky rabbit's foot with him each time. Each session consists of 20 spins of the wheel. John puts \$50 on red at each spin. So he pays out \$1000 during the session and gets back \$100$R,$where$R$is the number of times Red occurs. Then at each session he wins$W = 100R - 1000$dollars. In R statistical software, the 100 sessions can be simulated as below (where numbers in brackets show the index of the first result in each row): set.seed(611) r = rbinom(100, 20, 18/38) w = 100*r - 1000 w [1] -200 0 100 0 -100 200 -200 -100 -300 -300 [11] -500 -100 -100 -200 -500 -100 -100 -300 -200 -100 [21] 300 -200 -300 -100 -100 100 -100 0 -100 0 [31] 0 400 100 -100 -300 0 -200 0 -100 -300 [41] -400 200 -400 -100 -300 0 -100 100 -200 -300 [51] -100 100 -200 0 -400 200 -100 -200 -500 -200 [61] -100 -200 400 100 -200 -200 -200 100 0 200 [71] -200 -300 -100 100 100 -200 0 -100 -300 0 [81] 100 -200 200 200 -100 400 0 -700 0 100 [91] 100 100 100 500 -400 200 -200 -100 -300 -200  John might brag about winning or having a good time (and a few drinks) and coming even or better in 40 of the 100 sessions, also that he won \$300 or more during 12 of the sessions. But the histogram seems to show that the 'house edge' puts him at an overall disadvantage. (Overall, during the 100 sessions he has lost a total of \\$8300.)

The data are reasonably close to normally distributed so we can use a one-sample t test to test the null hypothesis that the average winnings at the wheel are zero or negative (system not working) against the alternative that average winnings are positive (rabbit's foot works).

t.test(w, alte="gr")

One Sample t-test

data:  w
t = -3.9365, df = 99, p-value = 0.9999
alternative hypothesis: true mean is greater than 0
95 percent confidence interval:
-118.0085       Inf
sample estimates:
mean of x
-83


The large P-value shows that the null hypothesis cannot be rejected.

Some such test of a hypothesis based on a substantial amount of carefully collected data would be a way to assess a gambler's claim.

Note: If you have ready access to R, you could repeat exactly the same simulation I did by beginning the program with set.seed(611), chosen because today's date is 6/11. If you omit the set.seed statement, you will get a somewhat different simulation (but not likely one that shows that the rabbit's foot works).