While Mark was out playing poker last Saturday night, I worked at getting the database of his all-in hands current. Adding in his Saturday results, iIt stands at 1191 hands with usable data (a few had errors that meant I could not compute a probability). Out of those 1191 all-in hands, he has an expected mean of 0.573. His actual computed mean is 0.559 for a difference of only 0.014 overall.

Under the assumption of random chance, this is not a big deviation. The p-value of this result is .15. Not low enough to reject the null hypothesis of random chance.

On the other hand, as an attempt to disprove the theory that Mark’s luck at cards is worse than average, it doesn’t do so hot. These results are very consistent with the null hypothesis of “Mark’s luck is worse than average.”

**A new data collection effort**

Recently he’s started recording three particular hands: {5,2}, {Q, 8}, and {A, K}. He started after noticing what seems like an awful lot of five/deuce hands. Each individual hand of 2 cards are all equally likely under the assumption of random chance. They are not equally profitable when playing poker. The following table gives a value for the hand in Texas Hold’em and Mark’s frequency of each of these three hands for mark over his last four poker nights.

Poker Hand |
Nominal Value |
Game 8/31/13 |
Game 9/7/13 |
Game 9/21/13 |
Game 10/5/13 |
Total Frequency |

Ace/King |
21.5 |
0 |
1 |
0 |
1 |
1 |

Queen/Eight |
10 |
2 |
3 |
0 |
2 |
5 |

Five/Deuce |
5 |
15 |
5 |
1 |
2 |
21 |

*This distribution borders on spooky! *

The chi-squared result for the frequency against a uniform frequency is 0.0000126. This is low enough to reject the null hypothesis that the frequency is uniform.

Adding in the relative values of the hands, the results are consistent with the null hypothesis of “Mark’s luck is worse than average” and no where close to “Mark’s luck is average” – i.e. fits the uniform distribution.

**Pair Versus Pair Analysis**

Mark felt that he was losing more often than he ought when he held a higher pair versus pair contests. So I scrubbed the data down to the relevant hands and took a look at his win rate under those circumstances. I limited the contests to preflop all-in calls.

Out of a total of 1189 poker all in hands, there were 74 that met the criteria of being pair versus pair all-in before the flop. One of those hands was a tie – both players held a pocket pair of kings, which I removed from the analysis, leaving 73 hands for the analysis.

We expect to win approximately 1 out of 5 hands with the lower pair and 4 out of 5 hands when holding the upper pair.

Results were as follows:

Pair vs Pair
Hand |
Total Number Hands |
Expected Probability of Win |
Expected Number Wins: |
Actual Number Wins |
Actual % wins: |
Actual − Expected |
Binomial Probability Of Difference |

Upper Pair |
42 |
0.7675 |
32.24 |
29 |
0.6905 |
−3.24 |
0.16 |

Lower Pair |
31 |
0.1822 |
5.65 |
5 |
0.1613 |
−0.65 |
0.49 |

Combined |
73 |
0.5189 |
37.88 |
34 |
0.4658 |
−3.88 |
0.21 |

Not exactly astoundingly bad – it’s still consistent with random chance – but definitely below average. We can expect 21% of such samples of 73 pair or pair hands to have 34 or fewer wins under the null hypothesis of randomness. However, these results also are consistent with the null hypothesis of “Mark’s luck is worse than average.”