Question

THE TREAD LIVES OF THE SUPER TITAN TIRES UNDER NORMAL DRIVING CONDITIONS ARE NORMALLY DISTRIBUTED WITH A MEAN OF 40,000 MILES AND A STANDARD DEVIATION OF 2,000 MILESA. WHAT IS THE PROBABILITY OF THE TIRE SELECTED AT RANDOM WILL HAVE A TREAD LIFE OF MORE THEN 35,000 MILES B. IF FOUR NEW TIRES ARE INSTALLED IN A CAR AND THEY EXPIERENCE EVEN WEAR DETIERMINE THE PROBABILITY THAT ALL FOUR TIRES STILL HAVE USEFUL TREAD AFTER 35,000 OF DRIVING?

Answer 1

B = 97% ROUNDED DOWN.

Answer 2

Step-by-step explanation

From the statement, we have a normal distribution for the life expectancy of tires with:

• variable X = life expectancy in miles,

,

• mean μ = 40,000 ml,

,

• standard deviation σ = 2,000 ml.

A) We must compute the probability:

`P(X>35,000).`

We calculate the z-score for this probability:

`z=(x-\mu)/(\sigma)=(35,000-40,000)/(2,000)=-2.5.`

Using a table for z-scores, we find that:

`P(X>35,000)=P(Z>-2.5)=0.99379=99.379\%.`

So a tire selected at random has a 99.379% probability of having a tread life of more than 35,000 miles.

B) Now, if four new tires are installed in a car, we want to know the probability that all four tires are still useful tread after 35,000 of driving.

From question A, we know that each wire has a probability of Pₐ = 0.99373 of having a tread life of more than 35,000 miles. We assume that the tread lives of the tires are independent of each other. So the total probability that the four wires will be useful after 35,000 of driving is:

`P_B=P_A\cdot P_A\cdot P_A\cdot P_A=(P_A)^4=(0.99373)^4\cong0.97515\cong97.515\%.`Answer

A) 99.379%

B) 97.515%