An interstate highway has a maximum speed limit of 70 mph an

An interstate highway has a maximum speed limit of 70 mph and a minimum speed limit of 45 mph. Jennifer drives at an average speed of 80 mph while Rick drives at an average speed of 40 mph. Both of their speeds are normally distributed with the same standard deviation of 5 mph.

(a) How likely Jennifer will receive a ticket for driving too fast?

(b) How likely Rick will receive a ticket for driving too slow?

(c) Suppose they carpooled yesterday and flipped a fair coin to decide who will drive. If they received a ticket, how likely is it that Rick drove? Here we only need to consider the possivilities that Jennifer received a ticket for driving too slow and Rick received a ticket for driving too slow.

Solution

a)
Mean ( u ) =80
Standard Deviation ( sd )=5
Normal Distribution = Z= X- u / sd ~ N(0,1)  
P(X > 70) = (70-80)/5
= -10/5 = -2
= P ( Z >-2) From Standard Normal Table
= 0.9772                  

b)
Mean ( u ) =40
Standard Deviation ( sd )=5
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
P(X < 45) = (45-40)/5
= 5/5= 1
= P ( Z <1) From Standard Normal Table
= 0.8413                  

c)
P( that flip car decide Rick drove) = 0.50
From (b) we know that Rick will raise the ticket = 0.8413
P( likely is it that Rick drove ) = 0.5 * 0.8413 = 0.42065