# Answer to Question #371 in Statistics and Probability for mary

Question #371
The deck of cards (36 pieces) is randomly divided into 2 equal packs. What is the probability that: a) there will be two aces in each pack; b) there will be all four aces in one pack; c) there will be equal amount of red cards in each pack?
1
2010-07-06T11:03:15-0400
The number of ways to select 18 cards (half of the deck) out of 36 is: N = (С36)^18 = 36!/(18!*18!).

a) Total amount of aces in the deck is 4, the amount of &quot;not aces&quot; is 32.
The &quot;favorable&quot; case is when one half of the deck (18 cards) has got two aces and 16 &quot;not aces&quot;.
The number of ways that you can choose 2 out of 4 aces is (C4)^2 = 4!/(2!*2!).
The number of ways that you can select the 16 &quot;not aces&quot; out of 32 cards is (C32)^16 = 32!/(16!*16!).
The number of ways you can achieve &quot;favorable&quot; cards is: m = (C4)^2*(C32)^16 = 4!*32!/(2!*2!*16!*16!).
The probability of &ldquo;favorable&rdquo; cards is: P = m/N = ((C4)^2*(C32)^16)/(С36)^18 = 4!*32!*18!*18!/(2!*2!*16!*16!*36!) &asymp; 0.795.

b) The &quot;favorable&quot; case is when one half of the deck (18 cards) has got 4 aces and 14 &quot;not aces&quot;.
The number of ways that you can choose all 4 aces is (C4)^4=1.
The number of ways that you can select the 14 &quot;not aces&quot; out of 32 cards is (C32)^14=32!/(14!*18!).
The number of ways you can achieve &quot;favorable&quot; cards is: m = (C4)^4*(C32)^14 = 1*32!/(14!*18!).
The probability of &ldquo;favorable&rdquo; cards is: P = m/N = ((C4)^4*(C32)^14)/(С36)^18 = 1*32!*18!*18!/(14!*18!*36!) &asymp; 0.052.

c) Total amount of red cards in the deck is 18, the amount of black cards is also 18.
The &quot;favorable&quot; case is when one half of the deck (18 cards) has got 9 red and 9 black cards.
The number of ways that you can choose 9 out of 18 red cards is (C18)^9=18!/(9!*9!).
The number of ways that you can choose 9 out of 18 black cards is (C18)^9=18!/(9!*9!).
The number of ways you can achieve &quot;favorable&quot; cards is: m = (C18)^9*(C18)^9 = 18!*18!/(9!*9!*9!*9!).
The probability of &ldquo;favorable&rdquo; cards is: P = m/N = ((C18)^9*(C18)^9)/(С36)^18 = 18!*18!* 18!*18!/ (9!*9!*9!*9!*36!) &asymp; 0.26.

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