Answer: The probability that a randomly selected tire will have a tread-life of less than 65,000 miles is 0.7872 .
Step-by-step explanation:
The cumulative distribution function for exponential distribution is :-
[tex]P(X\leq x)=1-e^{\frac{-x}{\lambda}}[/tex], where [tex]\lambda[/tex] is the mean of the distribution.
As per given , we have
Average tread-life of a certain brand of tire : Â [tex]\lambda=\text{42,000 miles }[/tex]
Now , the probability that a randomly selected tire will have a tread-life of less than 65,000 miles will be :
[tex]P(X\leq 65000)=1-e^{\frac{-65000}{42000}}\\\\=1-e^{-1.54761}\\\\=1-0.212755853158\\\\=0.787244146842\approx0.7872[/tex]
Hence , the probability that a randomly selected tire will have a tread-life of less than 65,000 miles is 0.7872 .
