In: Statistics and Probability
The distance between major cracks in a highway follows an exponential distribution with a mean of 21 miles. Given that there are no cracks in the first five miles inspected, what is the probability that there is no major crack in the next 5 mile stretch? Please enter the answer to 3 decimal places.
SOLUTION:
From given data,
The distance between major cracks in a highway follows an exponential distribution with a mean of 21 miles. Given that there are no cracks in the first five miles inspected, what is the probability that there is no major crack in the next 5 mile stretch?
From the information, observe that the distance between major crackers in a highway follows an exponential distribution with mean of 21 miles .
Consider X is the random variable that represents the distance between major cracks in a highway.
The random variable X follows an exponential distribution
The density function is as follows
= >0
The distribution function is as follows:
F(x) = 1 -
The mean number of miles is, =21.
Calculate parameter
= 1 / = 1/21 = 0.047
Given that there are no cracks in the first five miles inspected, what is the probability that there is no major crack in the next 5 mile stretch?
Calculate the probability that there are no major cracks in the next 5 miles inspected given that there are no cracks in the first five miles inspected.
P(X >10 | X>5) = P(X > 5+5 | X>5)
= P(X > 5)
= 1 - P(X < 5)
= 1 - [1-]
=
= 0.79057
0.791 (answer to 3 decimal places)
Therefore, the probability that there are no major cracks in the next 5 miles inspected given that there are no cracks in the first five miles stretch is 0.791