A traffic light at a certain intersection is green 50% of the time, yellow 10% of the time, and red 40% of the time. A car approaches this intersection once each day. Let X represent the number of days that pass up to and including the first time the car encounters a red light. Assume that each day represents an independent trial.

a. Find P(X = 3).

b. Find P(X<3).

c. Find μX

d. Find

Answer:

Step 1 of 5:

Given, at a certain intersection of a traffic light, green is 50% of the time, yellow is 10% of the time and red is 40% of the time.

Let X = number of days that pass up to and including the first time the car encounters a red light.

Assumption is that each day represents an independent trial.

Let X follows Geometric distribution and probability mass function of the Geometric distribution is

P(x) = p (1 - p , x = 1, 2, …..

Where, p = 0.40

Step 2 of 5:

The claim is to find the P(x = 3)Then, P( x=3 ) = (0.4) (1 - 0.4

= (0.4) (0.6

= (0.4) (0.36)

= 0.144

Hence, P(x = 3) = 0.144

Step 3 of 5:

The claim is to find the P(x < 3)Then, P( x<3 ) = P(x=1) + P(x=2)

= (0.4) (1 - 0.4+(0.4) (1 - 0.4

= (0.4) (0.6 + (0.4) (0.6

= (0.4) (1) + (0.4) (0.6)

= 0.4 + 0.24

Hence, P(x < 3) = 0.64