Question

a standard deck of cards contains 52 cards. one card is selected from the deck
(a) compute the probability of randomly selecting a diamond or club
(b) compute the probability of randomly selecting a diamond or club or heart
(c) compute the probability of randomly selecting a queen or spade

(a) p(diamond or club) =
(type an integer or a decimal rounded to three decimal places as needed.)
(b) p(diamond or club or heart) =
(type an integer or a decimal rounded to three decimal places as needed.)
(c) p(queen or spade) =
(type an integer or a decimal rounded to three decimal places as needed.)

Explanation:

Step1: Calculate probability for (a)

Use the formula \(P(A\cup B)=P(A)+P(B)\) for mutually - exclusive events.
Number of diamonds \(n(D)=13\), number of clubs \(n(C)=13\).
\(P(D\cup C)=\frac{n(D)+n(C)}{n(S)}=\frac{13 + 13}{52}=\frac{26}{52}=0.5\)

Step2: Calculate probability for (b)

Number of diamonds \(n(D)=13\), number of clubs \(n(C)=13\), number of hearts \(n(H)=13\)
\(P(D\cup C\cup H)=\frac{n(D)+n(C)+n(H)}{n(S)}=\frac{13+13 + 13}{52}=\frac{39}{52}=0.75\)

Step3: Calculate probability for (c)

Use the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
Number of queens \(n(Q)=4\), number of spades \(n(S)=13\), number of queen of spades \(n(Q\cap S)=1\)
\(P(Q\cup S)=\frac{n(Q)+n(S)-n(Q\cap S)}{n(S)}=\frac{4 + 13-1}{52}=\frac{16}{52}\approx0.31\)

Answer:

(a) \(0.5\)
(b) \(0.75\)
(c) \(0.31\)