Question

calculating probabilities with a deck of cards
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: Determine total - outcomes formula

When choosing 5 - card hands from a 52 - card deck, the order of the cards in the hand does not matter. The combination formula \(_{n}C_{r}=\frac{n!}{r!(n - r)!}\), where \(n = 52\) (total number of cards) and \(r = 5\) (number of cards in the hand) is used. So the total possible outcomes are \(_{52}C_{5}=\frac{52!}{5!(52 - 5)!}\), and the permutation formula \(_{n}P_{r}=\frac{n!}{(n - r)!}\) is not applicable here since order doesn't matter.

Step2: Calculate 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 - card hands with 2 diamonds and 3 hearts is \(_{13}C_{2}\times_{13}C_{3}=78\times286 = 22308\). The total number of 5 - card hands is \(_{52}C_{5}=2598960\). The probability \(P=\frac{_{13}C_{2}\times_{13}C_{3}}{_{52}C_{5}}=\frac{22308}{2598960}\approx0.0086
eq0.089\).

Step3: Calculate probability of five - spades

The number of ways to choose 5 spades out of 13 is \(_{13}C_{5}=\frac{13!}{5!(13 - 5)!}=\frac{13\times12\times11\times10\times9}{5\times4\times3\times2\times1}=1287\). The probability \(P=\frac{_{13}C_{5}}{_{52}C_{5}}=\frac{1287}{2598960}\approx0.000495\approx0.0005\).

Step4: Calculate 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 \(P=\frac{_{13}C_{5}}{_{52}C_{5}}=\frac{1287}{2598960}\approx0.000495\approx0.0005\).

Answer:

The total possible outcomes can be found using \(_{52}C_{5}\).
The probability of choosing five clubs is roughly 0.0005.