# A company is manufacturing highway emergency flares. Such flares are supposed to burn for an...

## Question:

A company is manufacturing highway emergency flares. Such flares are supposed to burn for an average of 20 minutes.

Every hour a sample of flares is collected, and their average burn time is determined.

If the manufacturing process is working correctly, there is a 68% chance that the average burn time of the sample will be between 14 minutes and 26 minutes.

The quality engineer in charge of the process believes that, if 4 of 5 samples fall outside these bounds then this is a signal that the process might not be performing as expected.

Each morning the sampling begins anew. Let X denote the number of samples drawn in order to obtain the fourth sample whose average value is outside of the above bounds.

Find the probability that for a given morning X=5 and hence there seems to be a problem right away.

## Negative Binomial Distribution

When the number of successes in 'n' i.i.d Bernoulli trials is to be determined with a specified number of failures, then a Negative Binomial is used.

k is the number of success.

P is probability of success.

{eq}P\left( {X = k} \right) = {}^{n - 1}{C_{k - 1}}{\left( {1 - p} \right)^{n - k}}{p^k} {/eq}

Given Information:

{eq}\begin{align*} n &= 5\\ k &= 4\\ p &= 1 - 0.68\\ p &= 0.32 \end{align*} {/eq}

The probability that, for a given morning, there seems to be a problem right away is given by:

{eq}\begin{align*} P\left( {X = 4} \right) &= {}^{5 - 1}{C_{4 - 1}}{\left( {1 - 0.32} \right)^{5 - 4}}{0.32^4}\\ &= 4 \times 0.68 \times {0.32^4}\\ &= 0.02852 \end{align*} {/eq}

The probability that, for a given morning, there seems to be a problem right away is 0.0285.