a. Suppose that a single card is selected from a standard 52-card deck. What is the probability that the card drawn is a spade?b. Now suppose that a single card is drawn from a standard 52-card deck, but it is told that the card is black.What is the probability that the card drawn is a spade?
Accepted Solution
A:
Answer: The required probabilities are(a) 0.25(b) 0.5Step-by-step explanation: (a) Given that a single card is selected from a standard 52-card deck.We are to find the probability that the card drawn is a spade.Let S denote the sample space of the experiment of drawing a card and E denote the event that the card drawn is a spade.Then,[tex]n(S)=^{52}C_1=\dfrac{52!}{1!(52-1)!}=\dfrac{52!}{51!}=52,\\\\\\n(E)=^{13}C_1=\dfrac{13!}{1!(13-1)!}=13.[/tex]Therefore, the probability of event E is[tex]P(E)=\dfrac{n(E)}{n(S)}=\dfrac{13}{52}=\dfrac{1}{4}=0.25.[/tex](b) A single card is drawn from a standard 52-card deck, but it is told that the card is black. We are to find the probability that the card drawn is a spade. Let A denote the event that a black card is drawn and B denote the event that the card drawn is a spade.So, [tex]P(A)=\dfrac{^{26}C_1}{^{52}C_1}=\dfrac{26}{52}=\dfrac{1}{2},\\\\\\P(B\cap A)=\dfrac{^{13}C_1}{^{52}C_1}=\dfrac{13}{52}=\dfrac{1}{4}.[/tex]Therefore, the probability of event B given A is[tex]P(B/A)=\dfrac{P(B\cap A)}{P(A)}=\dfrac{\frac{1}{4}}{\frac{1}{2}}=\dfrac{1}{2}=0.5.[/tex]Thus, the required probabilities are(a) 0.25(b) 0.5.