Question

a standard deck of 52 playing cards contains 13 cards in each of four suits: diamonds, hearts, clubs, and spades. two cards are chosen from the deck at random.
what is the approximate probability of choosing one club and one heart?
0.0588
0.0637
0.1176
0.1275

Explanation:

Step1: Calculate the number of ways to choose 2 cards from 52

The formula for combinations is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 52\) and \(r=2\).

$$C(52,2)=\frac{52!}{2!(52 - 2)!}=\frac{52\times51}{2\times1}=1326$$

Step2: Calculate the number of ways to choose 1 club and 1 heart

There are 13 clubs and 13 hearts. The number of ways to choose 1 club out of 13 is \(C(13,1)=\frac{13!}{1!(13 - 1)!}=13\), and the number of ways to choose 1 heart out of 13 is \(C(13,1) = 13\). By the multiplication principle, the number of ways to choose 1 club and 1 heart is \(13\times13=169\)

Step3: Calculate the probability

The probability \(P=\frac{\text{Number of favorable outcomes}}{\text{Number of total outcomes}}\)

$$P=\frac{169}{1326}\approx0.1275$$

Answer:

0.1275