Hector and whirlen are racing each other head to head in a marathon. when hector runs a marathon, the probability he finishes within three hours is 0.42. what whirlen runs, the probability he finishes within three hours is 0.35. finishes within three hours, the conditional probability that hector also finishes within three hours is 0.50. p(h) = 0.42, p(w) = 0.36. assume also: given that whirlen let's use the notation: a) (3pts) what is the probability that both of them finish within three hours? b) (3pts) what is the probability that at least one of them finishes within 3 hours? c) (3pts) given that hector finishes within three hours, what is the conditional probability that whirlen also finishes within 3 hours?

a)   P(b)  = 0.147     or   14.7 %

b)    P(i)  =  0.77      or   77 %

c)    P [ A ║ B ]   = 0.6   or   60 %

Step-by-step explanation:

a) The events Hector runs a marathon and finished within three hours is totally independent of probabilities of Whirlen

then probability of both of them finishing within three hours is

P(b)  =  Probability of Hector (finishing within 3 hours) * Probability of Whirlen  (finishing within 3 hours )

P(b)  =  0.42 * 0.35     ⇒   P(b)  = 0.147      or      P(b)  = 14.7 %

b)  The probability of at least one of them finishes within three hours is

P(i)  =  Probability of Hector (finishing within 3 hours) + Probability of Whirlen (finishing within 3 hours)

P(i)  = 0.42  +  0.35       ⇒     P(i)  =  0.77      or     P(i)  = 77 %

c) The probability of whirlen finishes within 3 hours given that Hector finished within three hours is express according to Bayes Theorem

Event  A Hector finished within three hours  :  0.42

Event B  Whirlen finished within three hours  : 0.35

Bayes Theorem:

P [ A ║ B ]   =   P(A) *  P [ B ║ A ]  / P(B)

P [ A ║ B ]   = 0.42 * 0.50 / 0.35     ⇒  P [ A ║ B ]   = 0.6