Question

Critical-part failures in NASCAR vehicles. In NASCAR races such as the Daytona 500, 43 drivers start the race; however, about 10% of the cars do not finish due to the failure of critical parts. University of Portland professors conducted a study of critical-part failures from 36 NASCAR races (The Sport Journal, Winter 2007). The researchers discovered that the time (in hours) until the first critical-part failure is exponentially distributed with a mean of .10 hours.

a. Find the probability that the time until the first critical part failure is 1 hour or more.

b. Find the probability that the time until the first critical part failure is less than 30 minutes.

Short answer

a. The probability that the time until the first critical part failure is more than an hour is 0.000045.

b. The probability that the time until the first critical part failure is less than 30 minutes is 0.9933

Step-by-step solution

Given Information

The time until the first critical part failure is exponentially distributed with a mean of 0.1hour

Let x be the time until the first critical part failure.

The p.d.f of the exponential distribution is given by

$f\left(x\right)=\frac{1}{\theta }{e}^{-\frac{1}{\theta }x};x>0$

(a) Compute the probabilities for given conditions

The probability that the time until the first critical part failure is more than an hour is computed as

$\begin{array}{rcl}p\left(x>1\right)& =& 10\underset{1}{\overset{\infty }{\int }}{e}^{-10x}dx\\ & =& {\left.\frac{10\times e^{-10x}}{-10}\right]}_1^{\infty }\\ & =& -e^{-10\infty }+e^{-10}\\ & =& 0.000045\end{array}$

Thus, the probability is 0.000045

(b) Compute the probability

The probability that the time until the first critical part failure is less than 30 minutes is computed as

Therefore,

$\begin{array}{rcl}p\left(x<\frac{1}{2}\right)& =& 10\underset{0}{\overset{\frac{1}{2}}{\int }}{e}^{-10x}dx\\ & =& {\left.\frac{10\times e^{-10x}}{-10}\right]}_0^{\frac{1}{2}}\\ & =& -e^{-5}+e^{-0}\\ & =& 1-0.0067\\ & =& 0.9933\end{array}$

Thus, the probability is 0.9933.