The route used by a certain motorist in commuting to work contains two intersect

Question

The route used by a certain motorist in commuting to work contains two intersections with traffic signals. The probability that he must stop at the first signal is 0.45, the analogous probability for the second signal is 0.55, and the probability that he must stop at at least one of the two signals is 0.6. (a) What is the probability that he must stop at both signals? (b) What is the probability that he must stop at the first signal but not at the second one? (c) What is the probability that he must stop at exactly one signal? Need Help?Read It Talk to & Tutor

Explanation / Answer

Solution:

Given:

P( Stop at the first signal) = 0.45

P( Stop at the second signal)= 0.55

P( Stop at at least one of the two signals)= 0.6

Let A = Stop at the first signal and B = Stop at the second signal

Then,

P(A) = 0.45 , P(B) = 0.55 and P( A or B) = 0.6

Part a) Find P( Stop at both signals) = ...?

that is:

P( A and B) = ....?

Using the addition rule of probability:

P( A or B) = P(A) + P(B) - P( A and B)

0.6 =   0.45 + 0.55 - P(A and B)

0.6 =1.00 - P(A and B)

Thus

P(A and B) = 1.00 - 0.6

P(A and B) = 0.4

Thus P( He must stop at both signals) = 0.4

Part b) P( He must stop at the first signal but not at the second one)=........?

that is: P( A and Bc) = ......?

P( A and Bc) = P(A) - P( A and B)

P( A and Bc) = 0.45 - 0.4

P( A and Bc) = 0.05

Thus

P( He must stop at the first signal but not at the second one)= 0.05

Part c) P( He must stop at exactly one signal) = ......?

that is: P( A or B) - P(A and B) = .....?

P( A or B) - P(A and B) = 0.6 - 0.4

P( A or B) - P(A and B) = 0.2

Thus P( He must stop at exactly one signal) = 0.2

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