How do you find the probability of drawing 5 red cards in 5 card poker hand?

Answer 1

A quick way is to simply count all the possible hands with red cards and compare to the total number of hands. There are 52 total cards and 26 red cards. We can use the binomial coefficient for this, otherwise known as the "choose" operator.

A = Total Hands: (52c5) = 2598960
B = Red Hands: (26c5) = 65780
C = Chance of red hand = A / B = 2.53%

Alternately you can use a permutation solution, for each card, in order, what is the chance to get a red card. There are 26 of 52 red cards on the first card, so 26/52. Then continue:

26/52 * 25/51 * 24/50 * 23/49 * 22/48 = 2.53%

If you play with the form of the equations, and understand a bit about combinations, you'll see that this second form is very closely related to the expanded version of the first form. This two methods have advantages in different situations. Here they are about the same.