Question

a deck of playing cards has four suits, with thirteen cards in each suit consisting of the numbers 2 through 10, a jack, a queen, a king, and an ace. the four suits are hearts, diamonds, spades, and clubs. a hand of five cards will be chosen at random. which statements are true? check all that apply. the total possible outcomes can be found using $_{52}c_{5}$. the total possible outcomes can be found using $_{52}p_{5}$. the probability of choosing two diamonds and three hearts is 0.089. the probability of choosing five spades is roughly 0.05. the probability of choosing five clubs is roughly 0.0005.

Explanation:

Step1: Combinations vs Permutations

When choosing a hand of five cards from a deck of 52 cards, the order of the cards in the hand does not matter. The formula for combinations \(_{n}C_{r}=\frac{n!}{r!(n - r)!}\) and for permutations \(_{n}P_{r}=\frac{n!}{(n - r)!}\). Since order doesn't matter for a card - hand, the total number of possible outcomes is \(_{52}C_{5}\), so the first statement is true and the second statement (\(_{52}P_{5}\)) is false.

Step2: Probability of two diamonds and three hearts

The number of ways to choose 2 diamonds out of 13 is \(_{13}C_{2}=\frac{13!}{2!(13 - 2)!}=\frac{13\times12}{2\times1}=78\). The number of ways to choose 3 hearts out of 13 is \(_{13}C_{3}=\frac{13!}{3!(13 - 3)!}=\frac{13\times12\times11}{3\times2\times1}=286\). The number of ways to choose 5 cards from 52 is \(_{52}C_{5}=\frac{52!}{5!(52 - 5)!}=2598960\). The probability \(P=\frac{_{13}C_{2}\times_{13}C_{3}}{_{52}C_{5}}=\frac{78\times286}{2598960}\approx0.0085
eq0.089\), so this statement is false.

Step3: Probability of five spades

The number of ways to choose 5 spades out of 13 is \(_{13}C_{5}=\frac{13!}{5!(13 - 5)!}=1287\). The probability \(P=\frac{_{13}C_{5}}{_{52}C_{5}}=\frac{1287}{2598960}\approx0.000495
eq0.05\), so this statement is false.

Step4: Probability of five clubs

The number of ways to choose 5 clubs out of 13 is \(_{13}C_{5}=\frac{13!}{5!(13 - 5)!}=1287\). The probability \(P = \frac{_{13}C_{5}}{_{52}C_{5}}=\frac{1287}{2598960}\approx0.0005\), so this statement is true.

Answer:

The first statement (The total possible outcomes can be found using \(_{52}C_{5}\)) and the fifth statement (The probability of choosing five clubs is roughly \(0.0005\)) are true.