Question

An ordinary deck of 52 playing cards is randomly divided into 4piles of 13 cards each. Compute the probability that each pile hasexactly 1 ace.

Hint : Multiplication rule
Define events Ei , i = 1,2,3,4 as follows:

E1= { The ace of spades is in any of the piles}
E2= { The ace of spades and the ace of hearts arein different piles}
E3= { The aces of spades, hearts, and diamonds are allin different piles}
E4= { All 4 aces are in different piles}

Answer 1

The probability that each of four piles has exactly one ace is 1/4, calculated using the independent probabilities that each ace ends up in a separate pile.

Step-by-step explanation:

To calculate the probability that each pile of cards has exactly one ace after dividing a well-shuffled standard deck of 52 cards into 4 piles of 13 cards each, we use the multiplication rule. Consider the following events: Event E1 represents the probability that the ace of spades is in any of the piles. Since it can go in any pile, this is a certain event, so P(E1) = 1. Event E2 is the probability that the ace of spades and the ace of hearts are in different piles. Since the ace of spades is already placed, the ace of hearts has 3 piles it can go to avoid the ace of spades, resulting in P(E2) = 3/4.

Next, Event E3 is the probability that the aces of spades, hearts, and diamonds are all in different piles. There are 2 out of the remaining 3 piles where the ace of diamonds can be placed without coinciding with the other two aces, so P(E3) = 2/3. Lastly, Event E4 is where all 4 aces are in different piles. The last ace, the ace of clubs, has only one pile left to go to, making P(E4) = 1/2. To find the overall probability, multiply the probabilities of the independent events E2, E3, and E4 together:

P(All aces in different piles) = P(E1) × P(E2) × P(E3) × P(E4) = 1 × (3/4) × (2/3) × (1/2) = 1/4.