**Question**

A soda bottling company’s manufacturing process is calibrated so that 99% of bottles are filled to within specifications, while 1% is not within specification. Every hour, 12 random bottles are taken from the assembly line and tested. If 2 or more bottles in the sample are not within specification, the assembly line is shut down for recalibration. What is the probability that the assembly line will be shut down, given that it is actually calibrated correctly? Use Excel to find the probability.

- Round your answer to three decimal places.

**Question**

Alex wants to test the reliability of “lie detector tests,” or polygraph tests. He performs a polygraph test on a random sample of 12 individuals. If there is more than a 50% chance that the tests result in no false positives (that is, the test does not result in a true statement being recorded as a lie), Alex will conclude that the tests are reliable. If the probability of a lie detector test resulting in a false positive is 5.5%, what will Alex conclude? Use Excel to find the probability, rounding to three decimal places.

Select the correct answer below:

Alex will conclude that the test is reliable since the probability of no false positives is less than 0.5.

Alex will conclude that the test is not reliable since the probability of no false positives is greater than 0.5.

Alex will conclude that the test is not reliable since the probability of no false positives is less than 0.5.

**Question**

A database system assigns a 32-character ID to each record, where each character is either a number from 0 to 9 or a letter from A to F. Assume that each number or letter being selected is equally likely. Find the probability that at least 20 characters in the ID are numbers. Use Excel to find the probability.

**Question**

A roulette wheel has 38 slots, numbered 1 to 36, with two additional green slots labelled 0 and 00. Jim, a dealer at a roulette table, tests the roulette wheel by spinning a ball around the wheel repeatedly and seeing where the ball lands. The ball has an equally likely chance of landing in each slot. If Jim spins the ball around the wheel 25 times, what is the probability that the ball lands in a green slot at most twice? Use Excel to find the probability.

**Solution:**

**Question**

A soda bottling company’s manufacturing process is calibrated so that 99% of bottles are filled to within specifications, while 1% is not within specification. Every hour, 12 random bottles are taken from the assembly line and tested. If 2 or more bottles in the sample are not within specification, the assembly line is shut down for recalibration. What is the probability that the assembly line will be shut down, given that it is actually calibrated correctly? Use Excel to find the probability.

- Round your answer to three decimal places.

*Answer*

0.006.

*Explanation: *

Note that this is a binomial probability. In this case, we want to find the probability of 2 or more successes, where a success is a bottle not filled to within specifications. This is the complement of the probability of 0 to 1 successes, inclusive. To determine the probability from a binomial distribution using Excel, follow the steps below.

- First press
**FORMULAS**and then**INSERT FUNCTION**. - Then select the
**BINOM.DIST**function. - Next enter the values for the number of successes, the number of trials, the probability of a success, and the number of successes. In this case, enter 1, 12, and 0.01, in that order. Enter 1for
**Cumulative**since this is a cumulative probability. - Press
**OK**. Excel should then display the probability. Here, the resulting probability is 0.993825, which is 0.994rounded to three decimal places.

To find the probability of 2 or more successes, subtract the probability of 0 or 1 successes from 1. This probability is 1−0.994=0.006.

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