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# Probability in Selecting Cards

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Suppose we randomly select 5 cards from an ordinary deck of playing cards. We might ask: What is the probability of selecting AT MOST 2 Aces? At least two Aces?

https://brainmass.com/statistics/probability/probability-selecting-cards-182200

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Let C(n, m) = n! / (m! * (n-m)!) be the number of ways to select m items from n items.

There are C(52, 5) ways to select 5 cards from an ordinary deck of playing cards (52 cards).
(1) To select at most 2 Aces, we could select 0 Ace, 1 Aces or 2 Aces.
There are C(46, 5) ways to select 0 Ace, C(5, 1) * C(46, 4) ways to select 1 Ace, C(5, 2) * C(46, 3) ways to select
2 Aces. Thus the probability is
(C(46, 5) + C(5, 1) * C(46, 4) + C(5, 2) * C(46, 3)) / C(52, 5) = 2338479 / 2598960 = 81.36%
(2) To select at least 2 Aces, we could select 2 Aces, 3 Aces or 4 Aces.
There are C(5, 2) * C(46, 3) ways to select 2 Aces. From (1), the probability is
1 - 81.36% + C(5, 2) * C(46, 3) / C(52, 5) = 24.48%

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