here is my specific question:
If two people draw independently without replacement, different numbers of cards from different (complete) decks. How do I figure out the expected number of matches or the intersection of the two vectors of cards?
I know that you can get the combinations 52 choose 10 and 52 choose 12, if the people drew 10 and 12 cards, respectively. But I'm having trouble figuring out how many exact matches (number and suit) we would expect to see in the set of 22 cards drawn.
For some context, I plan to use the expected number of matches in a chi squared test as the expected value to determine if my experimental results were due to chance.
Thanks in advance for your help.
Each of the ten cards has a chance of $\frac{12}{52}$ to match with one of the 12 cards of the other deck, so the expected value of matches is
$\frac{10*12}{52}=\frac{30}{13}$
For a chi-squared-test,you nedd the probability for 0,1,... matches to find a critical value.
For this, you can use the hypergeometric distribution.
The probability of k matches is
$$\frac{\binom{12}{k}*\binom{40}{10-k}}{\binom{52}{10}}$$
Since the probability to have 5 or more matches is less than 0.05, you can choose 5 as a critical value. If 5 or more matches occur, you reject the hypothesis of a random draw.
If you prefer probability 0.01 , then the critical value is 6.