The probability is .60 that a famous Nigerian distance runner will enter the Boston marathon. If he does not enter, the probability that last year's winner will repeat is 0.66, but if he enters, the probability that last year's winner will repeat is only .18. What is the probability that last year's winner will repeat?

Pat has asked a professor for a recommendation letter for graduate school. She estimates that the probability that the letter will be strong is 0.5 and that the probability that the letter will be weak is 0.2, and mediocre 0.3. Pat also estimates that if the letter is strong, the probability that she will enter grad school is 0.8; if it is weak, 0.1; and if it is mediocre, 0.4. Given that she did enter grad school, what is the probability that the recommendation letter was strong?

The probability distribution for the random variable x follows.
x------f(x)
20-----0.21
25-----0.11
32-----0.27
37-----0.41

1. f(x) is greater than 0 for all x. Is this a valid probability distribution?

2. What is the probability that x = 32?

3. What is the probability that x is less than or equal to 25?

4. What is the probability that x is greater than 32?

The probability that a randomly selected person has high blood pressure (the event H) is p(H)=0.2 and the probability that a randomly selected is a runner (the event R) is P(R)=0.4. The probability that a randomly selected person has blood pressure and is a runner is 0.1. Find the probability that a randomly selected person is a runner, given that he has high blood pressure.

(a) 0

(b) 0.25

(c) 1

(d) 0.50

(e) 0.17

(f) None of the above