Question

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​?

Answer 1

Answer:  The required probabilities are

(a) 0.25

(b) 0.5

Step-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,

$n(S)=^{52}C_1=\dfrac{52!}{1!(52-1)!}=\dfrac{52!}{51!}=52,\\\\\\n(E)=^{13}C_1=\dfrac{13!}{1!(13-1)!}=13.$

Therefore, the probability of event E is

$P(E)=\dfrac{n(E)}{n(S)}=\dfrac{13}{52}=\dfrac{1}{4}=0.25.$

(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,

$P(A)=\dfrac{^{26}C_1}{^{52}C_1}=\dfrac{26}{52}=\dfrac{1}{2},\\\\\\P(B\cap A)=\dfrac{^{13}C_1}{^{52}C_1}=\dfrac{13}{52}=\dfrac{1}{4}.$

Therefore, the probability of event B given A is

$P(B/A)=\dfrac{P(B\cap A)}{P(A)}=\dfrac{\frac{1}{4}}{\frac{1}{2}}=\dfrac{1}{2}=0.5.$

Thus, the required probabilities are

(a) 0.25

(b) 0.5.