calculating probabilities with a deck of cards\na deck of playing cards has four suits, with thirteen cards…

calculating probabilities with a deck of cards\na 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.\nwhich statements are true? check all that apply.\nthe total possible outcomes can be found using $_{52}c_{5}$.\nthe total possible outcomes can be found using $_{52}p_{5}$.\nthe probability of choosing two diamonds and three hearts is 0.089.\nthe probability of choosing five spades is roughly 0.05\nthe probability of choosing five clubs is roughly 0.0005.

calculating probabilities with a deck of cards\na 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.\nwhich statements are true? check all that apply.\nthe total possible outcomes can be found using $_{52}c_{5}$.\nthe total possible outcomes can be found using $_{52}p_{5}$.\nthe probability of choosing two diamonds and three hearts is 0.089.\nthe probability of choosing five spades is roughly 0.05\nthe probability of choosing five clubs is roughly 0.0005.

Answer

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\neq0.089).

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.