Exercise 4.6. Conditioning on cards. Draw one card from a shuffled deck of 52 ca

Question

Exercise 4.6. Conditioning on cards. Draw one card from a shuffled deck of 52 cards. a. What is the probability that the card is a spade if you know the card is b. What is the probability that the card is a spade if you know the card is c. What is the probability that the card is a 7 if you know the card is a d. What is the probability that the card is a 7 if you know the card is black? a 7? black? spade? e. What is the probability that the card is black if you know the card is a spade? f. What is the probability the card is black if you know the card is a 7?

Explanation / Answer

We know that the probability of A given B is: P(A/B) = P(AB)/P(B).

Let S = Spade; 7 = 7; B = Black;

a. Solution:

P(S/7) = P(S7)/P(7) = (1/52)/(4/52) = 1/4

[Note: P(S/7) means the probability of spade given 7 and (1/52)/(4/52) means 1/52 divided by 4/52)].

b. Solution:

P(S/B) = (13/52)/(26/52) = 1/2

c. Solution:

P(7/S) = P(7S)/P(S) = (1/52)/(13/52) = 1/13

d. Solution:

P(7/B) = P(7B)/P(B) = (2/52)/(26/52) = 1/13

e. Solution:

P(B/S) = P(BS)/P(S) = (13/52)/(13/52) = 1

f. Solution:

P(B/7) = P(B7)/P(7) = (2/52)/(4/52) = 1/2

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