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 0.5, and the probability that he must stop at at least one of the two signals is 0.9.

(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?

Explanation / Answer

Let us call the first signal A & the second signal B
Given that, P[A] = 0.45,

P[B] = 0.5,

P[A or B] = 0.9
a)
now we know that the addition rule of probability ,

P[A or B] = P[A] + P[B] - P[A & B]

0.9 = 0.45 + 0.5 - P[A & B]

so P[A & B] = 0.95 - 0.9

= P[ stopping at both signals ]

= 0.05

b)
P[ stopping only at the 1st signal ]
= P[A] - P[A & B]
= 0.45 - 0.05
= 0.4

c)
P[ stopping only at the 2nd signal ]
= P[B] - P[A & B]
= 0.5 - 0.05
= 0.45
we have already found P[stopping only at 1st signal]

adding the two, we get
P[stopping at only one signal]
= 0.4+ 0.45
= 0.85

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