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Chapter 8: Problem 6

Two cards are drawn from a well-shuffled deck of 52 playing cards. Let \(X\) denote the number of aces drawn. Find \(P(X=2)\)

Short Answer

Expert verified
The probability of drawing both aces from a well-shuffled deck of 52 playing cards is \(P(X=2) = \frac{1}{221}\).

Step by step solution

01

Identify Relevant Information

In this problem, we need to find the probability of drawing two aces out of a deck of 52 playing cards. There are 4 aces and a total of 52 cards in the deck.
02

Calculate the Number of Combinations

We need to find the number of ways to draw two aces out of four, which can be represented as C(4,2). To do this, we will use the combination formula: C(n,r) = \(\frac{n!}{r!(n-r)!}\) Where n is the total number of items, r is the number of items we want to choose, and ! denotes the factorial function.
03

Calculate C(4,2)

Using the combination formula, we will now calculate the number of ways to draw two aces out of four: C(4,2) = \(\frac{4!}{2!(4-2)!}\) Performing factorials: C(4,2) = \(\frac{24}{(2)(2)}\) C(4,2) = 6 There are 6 ways to draw two aces out of four.
04

Calculate the Total Number of Ways to Draw Two Cards

Now, we need to find the total number of ways to draw two cards out of 52, represented by C(52,2). Using the combination formula: C(52,2) = \(\frac{52!}{2!(52-2)!}\)
05

Calculate C(52,2)

Calculate the number of ways to draw two cards out of 52: C(52,2) = \(\frac{52!}{2!50!}\) C(52,2) = \(\frac{52*51*50!}{2!50!}\) After canceling the 50!: C(52,2) = \(\frac{52*51}{2}\) C(52,2) = 1326 There are 1326 ways to draw two cards out of 52.
06

Calculate the Probability

To find the probability of drawing two aces, divide the number of ways to draw two aces by the total number of ways to draw two cards: P(X=2) = \(\frac{6}{1326}\) Simplifying: P(X=2) = \(\frac{1}{221}\) So the probability of drawing both aces from a well-shuffled deck of 52 playing cards is \(\frac{1}{221}\).

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