The probability that a car is going over the speed limit on I-77 is 76%. Suppose

Question

The probability that a car is going over the speed limit on I-77 is 76%. Suppose a random sample of 10 cars is observed. Find the following:

a) What is the expected value for the number of cars going over the speed limit?

b) What is the standard deviation for the number of cars going over the speed limit?

c) What is the probability that exactly 8 cars are going over the speed limit?

d) What is the probability that none of the cars are going over the speed limit?

e) What is the probability that at least 1 cars is going over the speed limit?

Please show work.

Explanation / Answer

The probability that a car is going over the speed limit on I-77 is 76%
random sample of 10 cars is observed

a.
the expected value for the number of cars going over the speed limit = n*p =10*0.76 =7.6

b.
the standard deviation for the number of cars going over the speed limit =sqrt((n*p)*(1-p)) =sqrt(7.6(1-0.76))
standard deviation = 1.3506

c.
the probability that exactly 8 cars are going over the speed limit
P(X > 8) = (8-7.6)/1.3504
= 0.4/1.3504 = 0.2962
= P ( Z >0.2962) From Standard Normal Table
= 0.3835,
P(X < = 8) = (1 - P(X > 8)
= 1 - 0.3835 = 0.6165

d.
the probability that none of the cars are going over the speed limit
P( X = 0 ) = ( 10 0 ) * ( 0.76^0) * ( 1 - 0.76 )^10
= 0

e.
the probability that at least 1 cars is going over the speed limit
P(X < 1) = (1-7.6)/1.3504
= -6.6/1.3504= -4.8874
= P ( Z <-4.8874) From Standard Normal Table
= 0
P(X > = 1) = (1 - P(X < 1)
= 1 - 0 = 1

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