Question

Suppose at a particular traffic light junction, the traffic is green 60 % of the time, red
30 % of the time and yellow 10 % of the time. Given that a car approaches this traffic
light junction once each day. Let X denotes 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.

(i) Apply a suitable model and compute the probability P(X = 3).

(ii) Compute the probability P(X ≤ 4).

Answer 1

Let X denotes the number of days that pass up to and including the first time the car encounters a red light and the trials are independent

The model used can be the geometric, where p = 30% = 0.3

and q = 1 - p= 1 - 0.3 = 0.7

(i) Apply a suitable model and compute the probability P(X = 3)

Using geometric P[ X = x+1 ] = q^x*p

P(X = 3) = 0.7^2*0.3

P(X = 3) = 0.147

(ii) Compute the probability P(X ≤ 4)

P[ X <= 4 ] = P[ X = 1 ] + P[ X = 2 ] + P[ X = 3 ] + P[ X = 4 ]

P[ X <= 4 ] = 0.7^0*0.3 + 0.7^1*0.3 + 0.7^2*0.3 + 0.7^3*0.3

P[ X <= 4 ] = 0.3 + 0.21 + 0.147 + 0.1029

P[ X <= 4 ] = 0.7599

P[ X <= 4 ] = 0.76