You pick 13 cards out of a deck of 52 What is the probabilit

You pick 13 cards out of a deck of 52. What is the probability that you will not get a card from every suit?

Solution

I interpret \"will not get a card from every suit\" to mean \"at least one suit is missing from the hand\", i.e. in bridge terms you have one or more voids. Total hands = 52 C 13 Hands from just 3 suits: 39C13 * 4 choices of 3 suits Hands from 2 suits: (included in the above) 26C13 * 6 pairs of suits (subtract to avoid double counting) Hands from 1 suit: (included in both of the above) 4 (add back, since taken out in previous) (39C13 * 4 - 26C13 * 6 + 4) / 52C13 ( 8122425444 * 4 - 10 400 600 * 6 + 4 ) / 635013559600 32427298180 / 635013559600 = 0.051 Another Method: The probability of no spades is simply 39C13 / 52C13 = 0.0127908 Since there are 4 suits the overall probability is 4 * 0.0127908 = 0.051163792