Answer:
The question here is incomplete, the complete question will be:
A standard card deck consists of 52 cards divided into 4 suits of 13 cards each. The 13 cards (ranks) of each suit are 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A. Suppose a standard deck is well-shuffled and then divided into 13 piles of 4 cards each. Let the random variable X denote the number of piles that each have no two cards of equal rank. Calculate E[X] to 3 significant digits. (Notice that the 13 events corresponding to the 13 piles having distinct ranks are not independent events. This is easier to see if the deck consisted of only 16 cards and there were only 4 ranks. Then, if the first 3 piles each had 4 distinct ranks, the probability would be 1 that the last pile has 4 distinct ranks.)
The answer to this question will be:
13⁴ / (5 2 4) ≅ 0.1055.
Step-by-step explanation:
There are 52 places where an ace might be put (13 places in each of 4 piles). We don't care which ace is which, so we can count (524) different ways to choose the four places where the aces will be put; each of these sets of four places is equally likely.
But to get exactly one ace in each pile, we have to choose one of the 13 places in the first pile, one of the 13 places in the second pile, one of the 13 places in the third pile, and one of the 13 places in the fourth pile. There are 134 ways to do that, so 134 of the sets of four places satisfy the condition. The probability is therefore
13⁴ / (5 2 4) ≅ 0.1055.
