Respuesta :
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,
[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.
