A commuter must pass through 3 traffic lights on her way to

A commuter must pass through 3 traffic lights on her way to work. For each traffic light, the probability that it is green when she arrives at the intersection is 0.7. You may assume that the traffic lights and the days are independent. (5 pts each)

(b) For the next 5 commuting days, let X = number of days that all 3 lights are green.

What is the probability distribution of X?

(c) Find P(X = 2)

(d) Find the mean and variance of X.

Solution

commuter must pass through 3 traffic lights on her way to work. For each traffic light, the probability that it is green when she arrives at the intersection is 0.7. You may assume that the traffic lights and the days are independent. (5 pts each)

P(all 3 lights are green) = 0.7*0.7*0.7=0.343

(b) For the next 5 commuting days, let X = number of days that all 3 lights are green.

What is the probability distribution of X?

X is binomial with P=0.343

x              P

0              0.122413

1              0.319540

2              0.333645

3              0.174186

4              0.045469

5              0.004748

(c) Find P(X = 2)

P(X=2) =0.333645

(d) Find the mean and variance of X.

Expectation = np = 1.715

Variance = np(1 - p) = 1.126755

Standard deviation = 1.061487