Question

Let us select five cards at random and without replacement from an ordinary deck of playing cards. (a) Find the pmf of \(X\), the number of hearts in the five cards. (b) Determine \(P(X \leq 1)\).

Short answer

The pmf of \(X\), the number of hearts, is calculated using the combination formula \(P(X=x) = \frac{C(13, x) * C(39, 5-x)}{C(52, 5)}\), and for \(P(X \leq 1)\), sum the probabilities of 0 and 1 heart.

Step-by-step solution

Basics of Combinations

A combination is the number of ways to select items (without regard to order) from a larger group. In a standard deck of 52 cards, there are 13 hearts. So, when you select 5 cards, the number of hearts \(X\) can range from 0 to 5. Using the combination notation, this can be represented as \(C(n, k) = \frac{n!}{k!(n-k)!}\), where \(n\) is the total number of items, \(k\) is the number of items to select, and \(!\) denotes factorial.

Computing PMF

The pmf of \(X\), the number of hearts, is given by the number of ways to get \(X\) hearts and (5 - \(X\)) non-hearts divided by the total number of ways to select 5 cards from the deck. Computationally, this is represented as \(P(X=x) = \frac{C(13, x) * C(39, 5-x)}{C(52, 5)}\), for \(x = 0, 1, 2, 3, 4, 5\). Use this formula to find the pmf for \(X\).

Calculating \(P(X \leq 1)\)

The probability \(P(X \leq 1)\) is the sum of the probabilities of 0 and 1 heart. So \(P(X \leq 1) = P(X=0) + P(X=1)\). Use the pmf calculated in Step 2 for these calculations.