I'm playing contract bridge and the boards are dealt by a machine. Last time, one of the player suggested the boards are not dealt at random and I want to check that.
At contract bridge, each player receive 13 cards from a deck of 52 cards. It's possible to categorize the hand of a player by its pattern. For instance, a 5-4-3-1 hand is a hand with 5 cards in one suit, 4 cards in another, 3 cards in the third one and just 1 card in the last one. The probabilities to have a specific hand pattern if the distribution is fair are known. For a 5-4-3-1 hand, we have a probability of 12.931%.
Now, I want to design a protocol to check if the distribution is fair or not. My null hypothesis is "the distribution is fair" and my alternative hypothesis is "the distribution is unfair". My question is: is the following protocol sufficient to accept or reject my alternative hypothesis?
I want to use a Z-test for each hand pattern (39 different hand patterns). The idea is to count the frequency of each hand pattern over a sample of n different boards. Bonus question: what minimum value of n should I choose to have a 95% confidence interval?
The problem of the Z-test is I will have 39 different tests. What if some of them accept the alternative hypothesis and some others reject it? How can I sum the results to get a unique result? I would like to be able to say: "I'm sure the board distribution is fair at x percent."
Last point, this protocol just check the hand patterns but I want to do the same verification on the high cards points of each hand. Can I follow the same protocol? What if there are different results?