Question

1. avery has been learning to play some new card games and is curious about the probabilities of being dealt different cards from a standard 52 - card deck. help him figure out the probabilities listed below.

a. what are p(king), p(queen), and p(club)?
b. what is p(king or club)? how does your answer relate to the probabilities you calculated in part (a)?
c. what is p(king or queen)? again, how does your answer relate to the probabilities you calculated in part (a)?
d. what is the probability of not getting a face card? jacks, queens, and kings are face cards. that is, find p(not face card).

Explanation:

Step1: Calculate P(king)

There are 4 kings in a 52 - card deck. So $P(\text{king})=\frac{4}{52}=\frac{1}{13}$.

Step2: Calculate P(queen)

There are 4 queens in a 52 - card deck. So $P(\text{queen})=\frac{4}{52}=\frac{1}{13}$.

Step3: Calculate P(club)

There are 13 clubs in a 52 - card deck. So $P(\text{club})=\frac{13}{52}=\frac{1}{4}$.

Step4: Calculate P(king or club)

Use the formula $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Here $A$ is the event of getting a king and $B$ is the event of getting a club. $P(A) = \frac{4}{52}$, $P(B)=\frac{13}{52}$, and $P(A\cap B)=\frac{1}{52}$ (the king of clubs). So $P(\text{king or club})=\frac{4 + 13- 1}{52}=\frac{16}{52}=\frac{4}{13}$. It relates to part (a) as we used $P(\text{king})$ and $P(\text{club})$ in the formula.

Step5: Calculate P(king or queen)

Since the events of getting a king and getting a queen are mutually - exclusive, $P(\text{king or queen})=P(\text{king})+P(\text{queen})$. So $P(\text{king or queen})=\frac{4 + 4}{52}=\frac{8}{52}=\frac{2}{13}$. It relates to part (a) as we used $P(\text{king})$ and $P(\text{queen})$ directly.

Step6: Calculate P(not face card)

There are 12 face cards (4 jacks, 4 queens, 4 kings) in a 52 - card deck. So the number of non - face cards is $52 - 12=40$. Then $P(\text{not face card})=\frac{40}{52}=\frac{10}{13}$.

Answer:

a. $P(\text{king})=\frac{1}{13}$, $P(\text{queen})=\frac{1}{13}$, $P(\text{club})=\frac{1}{4}$
b. $P(\text{king or club})=\frac{4}{13}$
c. $P(\text{king or queen})=\frac{2}{13}$
d. $P(\text{not face card})=\frac{10}{13}$