Question

A traffic light at a certain intersection is green 45% of the time, yellow 10% of the time, and red 45% of the time. A car approaches this intersection once each day. Let Y denote the number of days up to and including the third day on which a red light is encountered. Assume that each day represents an independent trial.

Find ?_Y .

Answer 1

Since each day represents an independent trial, so there is 0.45 probability of seeing a red light every day.

Y=1 means once red light is encountered, which can happen on the 1st day, or second day, or third day.

For 1st day probability is 0.45, for second day it is (0.55)(0.45) and for 3rd day it is (0.55)(0.55)(0.45)

Y=2 means twice red light is encountered, which can happen on the first two days, or first and third day or second and third day. and so on

Hence by definition of mean, we get