a standard deck of 52 playing cards contains 13 cards in each of four suits: hearts, diamonds, clubs, and…

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?\n1%\n4%\n11%\n44%

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?\n1%\n4%\n11%\n44%

Answer

Explanation:

Step1: Calculate number of ways to choose 3 diamonds

We use combination formula $C(n,k)=\frac{n!}{k!(n - k)!}$. There are 13 diamonds, and we want to choose 3. So $C(13,3)=\frac{13!}{3!(13 - 3)!}=\frac{13\times12\times11}{3\times2\times1}=286$.

Step2: Calculate number of ways to choose 1 non - diamond

There are $52-13 = 39$ non - diamonds, and we want to choose 1. So $C(39,1)=\frac{39!}{1!(39 - 1)!}=39$.

Step3: Calculate total number of ways to choose 4 cards from 52

$C(52,4)=\frac{52!}{4!(52 - 4)!}=\frac{52\times51\times50\times49}{4\times3\times2\times1}=270725$.

Step4: Calculate probability

The number of ways to get exactly 3 diamonds is the product of the number of ways to choose 3 diamonds and 1 non - diamond, which is $286\times39 = 11154$. The probability $P=\frac{11154}{270725}\approx0.0412\approx4%$.

Answer:

4%