Question

a standard deck of cards contains 51 cards. one card is selected from the deck. compute the following probabilities: a. selecting a club or diamond. b. selecting a club or diamond or heart. c. selecting a five or spade.

Explanation:

Step1: Determine total number of cards

A standard deck has 52 cards (assuming there is a typo in the problem - should be 52 instead of 51). So, total number of outcomes $n(S)=52$.

Step2: Calculate probability for part a

Number of clubs $n(C)=13$, number of diamonds $n(D)=13$. Since clubs and diamonds are mutually - exclusive events, $n(C\cup D)=n(C)+n(D)=13 + 13=26$. Probability $P(C\cup D)=\frac{n(C\cup D)}{n(S)}=\frac{26}{52}=\frac{1}{2}$.

Step3: Calculate probability for part b

Number of clubs $n(C)=13$, number of diamonds $n(D)=13$, number of hearts $n(H)=13$. Since these are mutually - exclusive events, $n(C\cup D\cup H)=n(C)+n(D)+n(H)=13 + 13+13 = 39$. Probability $P(C\cup D\cup H)=\frac{n(C\cup D\cup H)}{n(S)}=\frac{39}{52}=\frac{3}{4}$.

Step4: Calculate probability for part c

Number of fives $n(5)=4$, number of spades $n(S_{p})=13$, and the five of spades is counted in both. So, $n(5\cup S_{p})=n(5)+n(S_{p})-n(5\cap S_{p})=4 + 13-1=16$. Probability $P(5\cup S_{p})=\frac{n(5\cup S_{p})}{n(S)}=\frac{16}{52}=\frac{4}{13}$.

Answer:

a. $\frac{1}{2}$
b. $\frac{3}{4}$
c. $\frac{4}{13}$