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 a hand of 5 cards from a deck of 52 cards, the order of the cards in the hand does not matter. The formula for combinations $_nC_r=\frac{n!}{r!(n - r)!}$, where $n = 52$ (total number of cards) and $r=5$ (number of cards to be chosen). So the total possible outcomes are given by $_{52}C_5$.

Step2: Calculate probability of two - diamonds and three - hearts

Number of diamonds $= 13$, number of ways to choose 2 diamonds is $_{13}C_2=\frac{13!}{2!(13 - 2)!}=\frac{13\times12}{2\times1}=78$. Number of hearts $=13$, number of ways to choose 3 hearts is $_{13}C_3=\frac{13!}{3!(13 - 3)!}=\frac{13\times12\times11}{3\times2\times1}=286$. The number of ways to get 2 diamonds and 3 hearts is $_{13}C_2\times_{13}C_3=78\times286 = 22308$. The total number of 5 - card hands is $_{52}C_5=\frac{52!}{5!(52 - 5)!}=2598960$. 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

Number of spades $= 13$, number of ways to choose 5 spades is $_{13}C_5=\frac{13!}{5!(13 - 5)!}=\frac{13\times12\times11\times10\times9}{5\times4\times3\times2\times1}=1287$. The probability $P=\frac{_{13}C_5}{_{52}C_5}=\frac{1287}{2598960}\approx0.0005
eq0.05$.

Step4: Calculate probability of five - clubs

Number of clubs $= 13$, number of ways to choose 5 clubs is $_{13}C_5=\frac{13!}{5!(13 - 5)!}=1287$. The probability $P=\frac{_{13}C_5}{_{52}C_5}=\frac{1287}{2598960}\approx0.0005$.

Answer:

The total possible outcomes can be found using $_{52}C_5$. The probability of choosing five clubs is roughly 0.0005. So the correct statements are:
The total possible outcomes can be found using $_{52}C_5$; The probability of choosing five clubs is roughly 0.0005.