A standard deck contains 52 cards, four of which are aces. If you are randomly dealt a five-card hand, what is the probability of having exactly three aces?

1 Answer
GM_Phrostbyte
Apr 9, 2018

The probability is #94/54145#.

Explanation:

Think of it as individual cards that makes up your hand. Note that the order you draw the cards in does not matter.

There are #\ _52C_5# combinations for a five-card hand, so the probability is the chosen/favourable outcomes divided by the total combinations.

Think of the next part this way: You can choose two cards that are not aces (there are 48), and that is simply #\ _48C_2#. This works because once you have the two cards that are not aces, you know that the other three will be aces.

Also, note that you have 4 different aces to choose from, and picking three of them results in more combinations, which is #\ _4C_3#.

Your final expression is #(\ _4C_3*\ _48C_2)/(\ _52C_5)#

Sorry that I do not have a general formula for this scenario.

The numerical answer is:

#(\ _4C_3*\ _48C_2)/(\ _52C_5)=(4*1128)/2598960#

#=94/54145#

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