Question

here is a table showing all 52 cards in a standard deck.
a five - card hand is dealt at random from a standard deck. (a five - card hand is any set of five different cards, chosen without replacement.)
what is the probability that the hand contains exactly two hearts?
round your answer to the nearest hundredth.

Explanation:

Step1: Calculate total number of 5 - card hands

The total number of ways to choose 5 cards from a deck of 52 cards is given by the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 52\) and \(r=5\). So, \(C(52,5)=\frac{52!}{5!(52 - 5)!}=\frac{52\times51\times50\times49\times48}{5\times4\times3\times2\times1}=2598960\).

Step2: Calculate number of ways to choose 2 hearts and 3 non - hearts

There are 13 hearts in a deck. The number of ways to choose 2 hearts is \(C(13,2)=\frac{13!}{2!(13 - 2)!}=\frac{13\times12}{2\times1} = 78\). There are \(52-13 = 39\) non - hearts. The number of ways to choose 3 non - hearts is \(C(39,3)=\frac{39!}{3!(39 - 3)!}=\frac{39\times38\times37}{3\times2\times1}=9139\).

Step3: Calculate number of 5 - card hands with exactly 2 hearts

By the multiplication principle, the number of 5 - card hands with exactly 2 hearts is \(C(13,2)\times C(39,3)=78\times9139 = 712842\).

Step4: Calculate the probability

The probability \(P\) is the number of favorable outcomes divided by the number of total outcomes. So, \(P=\frac{712842}{2598960}\approx0.27\).

Answer:

\(0.27\)