Probability that the track is dry if Red Bull teams is .935, if the Mclauren Tea

Question

Probability that the track is dry if Red Bull teams is .935, if the Mclauren Team wins is.915, and if the Ferrari Team wins is .844. The probability Red Bull Team, McLauren and Ferrari Team win are 385, .295 and .320, respectively. 4. a) b) c) d) List all probabilities given in this problem, expressed in terms of events that you define. What is the probability that the weather was dry? Given that track was dry, what is the probability that Redbull win? McLauren? Ferrari? Given that track was wet (not dry), what is the probability that Redbull win? Mclauren? Ferrari? is dry track condition independent of not being on the Ferrari team? e)

Explanation / Answer

a)

let R shows the event that Red Bull tem will win, M shows the event that McLauren team will win and F shwos the event that Ferrari team will win. From the given information we have

P(R) = 0.385, P(M) = 0.295, P(F) = 0.320

Let D shows the event that field is dry. So

P(D|R) = 0.935, P(D|M) = 0.915, P(D|F) = 0.844

b)

By the law of total probability, the probability that field is dry is

P(D) = P(D|R)P(R) + P(D|M)P(M) + P(D|F)P(F) = 0.935 * 0.385 + 0.915 * 0.295 + 0.844*0.320 = 0.359975 + 0.269925+ 0.27008 = 0.89998

c)

The required probabilites are

P(R|D) = [ P(D|R)P(R) ] / P(D) = 0.359975 / 0.89998 = 0.39998

P(M|D) = [ P(D|M)P(M) ] / P(D) = 0.269925 / 0.89998 = 0.29992

P(F|D) = [ P(D|F)P(F) ] / P(D) = 0.27008 / 0.89998 = 0.30010

d)

Let W shows the event field is wet so by the complement rule we have

P(W) = 1 -P(D) = 1- 0.89998 = 0.10002

and

P(W|R)=1-P(D|R) = 0.065, P(W|M)=1-P(D|M) = 0.085, P(W|F)=1-P(D|F) = 0.156

So,

P(R|W) = [P(W|R)P(R) ] /P(W) = [0.065 * 0.385] / 0.10002 = 0.2502

P(M|W) = [P(W|M)P(M) ] /P(W) = [0.085 * 0.295] / 0.10002 = 0.2507

P(F|W) = [P(W|F)P(F) ] /P(W) = [0.156 * 0.320] / 0.10002 = 0.4991

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