You have discovered the lights on the main street in Paris are synchronized, so

Question

You have discovered the lights on the main street in Paris are synchronized, so your
chances of getting a green light at the next intersection are higher if you have not been stopped
by a red light at the last intersection. You estimate that there is a 4/5 chance of getting a green
light if the previous light was green. Similarly, there’s only a 1/4 chance of getting a green light if
the last light was red. Because the main street in Paris is a major throughway, you
estimate there’s a 2/3 chance of the first light being green.
a) What is the probability that the second light will be green?
b) Out of the first three lights, how many lights can you expect to be green?
c) What is the probability of the third light being green when the first one was red?

Explanation / Answer

P( Green | Previous green ) = 0.8 because there is a 4/5 chance of getting a green light if the previous light was green.

Also, P( Green | Previous Red ) = 0.25 because there’s only a 1/4 chance of getting a green light if the last light was red.

P( Previous green ) = 2/3 therefore P( previous red ) = 1- (2/3) = 1/3

a) By addition law of probability we get:

P( Green ) = P( Green | Previous green ) *P( Previous green ) + P( Green | Previous Red ) P( previous red )

P( Green ) = 0.8*(2/3) + 0.25*(1/3) = 0.6167

Therefore the required probability here is 0.6167

b) Now probability that the second light is green is 0.6167, therefore probability that the second light is red is computed as: = 1 - 0.6167 = 0.3833

Now computing the probability of green light for the third light, we get the probability as:

P( Green third light ) = 0.6167*0.8 + 0.3833*0.25 = 0.5892

Expected number of greens out of the 3 lights is computed as:

= P(First light green ) + P(Second light green ) + P(Third light green )

= (2/3) + 0.6167 + 0.5892

= 1.8725

Therefore 1.8725 is the expected number of greens out of the first three lights.

c) Probability that the third light is green given that the first one was red, this can be in the following cases:

Therefore the required probability here is 0.2 + 0.1875 = 0.3875

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