Does a 55% win rate prove the desk has an edge?

Question

A desk places 400 independent directional bets that each either win or lose. It wins 220 of them, a 55% hit rate. The desk head wants to know whether this is evidence of a real edge or just luck.

Set up and carry out a formal test of "no edge" (win probability 0.5). Is 55% out of 400 statistically significant, and what caveat would you attach before anyone celebrates?

Accepted Answer

The hypotheses. Let p be the true win probability. Under "no edge," H0: p = 0.5; the two-sided alternative is H1: p 
eq 0.5. The observed proportion is \hat p = 220/400 = 0.55. The test statistic. Under H0, the number of wins is Binomial(400, 0.5), which the normal approximation handles well here. The standard error of \hat p under the null uses p = 0.5: 	ext{SE} = \sqrt{\frac{0.5 	imes 0.5}{400}} = \sqrt{0.000625} = 0.025. So z = \frac{\hat p - 0.5}{	ext{SE}} = \frac{0.55 - 0.50}{0.025} = 2.0. The p-value. A two-sided p-value is 2\,[1 - \Phi(2.0)] = 2 	imes 0.0228 \approx 0.046. That is…

Does a 55% win rate prove the desk has an edge?

Your answer

More Statistics questions