Question

a standard deck of 52 playing cards contains 13 cards in each of four suits: hearts, diamonds, clubs, and spades. four cards are drawn from the deck at random. what is the approximate probability that exactly three of the cards are diamonds? 1% 4% 11% 44%

Explanation:

Step1: Calculate total 4-card combinations

The total number of ways to choose 4 cards from 52 is given by the combination formula $C(n,k)=\frac{n!}{k!(n-k)!}$.
$$C(52,4)=\frac{52!}{4!(52-4)!}=270725$$

Step2: Calculate favorable combinations

We need 3 diamonds (from 13) and 1 non-diamond (from 39). Calculate each combination and multiply:
$$C(13,3)=\frac{13!}{3!(13-3)!}=286$$
$$C(39,1)=\frac{39!}{1!(39-1)!}=39$$
$$\text{Favorable combinations}=286\times39=11154$$

Step3: Compute probability

Divide favorable combinations by total combinations, then convert to percentage.
$$\text{Probability}=\frac{11154}{270725}\approx0.0412=4.12\%$$

Answer:

4%