Packaging Error
Because of a manufacturing error, three cans of regular soda were accidentally filled with diet soda and placed into a 12-pack. Suppose that two cans are randomly selected from the 12-pack.
c. Determine the probability that exactly one is diet and one is regular?

Verified step by step guidance

1. Identify the total number of cans and the composition: there are 12 cans in total, with 3 diet soda cans and 9 regular soda cans.
2. Determine the total number of ways to select 2 cans from the 12-pack. This is a combination problem calculated by the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), so compute \(\binom{12}{2}\).
3. Calculate the number of ways to select exactly one diet soda can and one regular soda can. This involves choosing 1 can from the 3 diet cans and 1 can from the 9 regular cans, so compute \(\binom{3}{1} \times \binom{9}{1}\).
4. Find the probability by dividing the number of favorable outcomes (one diet and one regular) by the total number of possible outcomes (any 2 cans from 12). The probability is given by \(\frac{\binom{3}{1} \times \binom{9}{1}}{\binom{12}{2}}\).
5. Interpret the result as the likelihood that when two cans are randomly selected from the 12-pack, exactly one will be diet soda and one will be regular soda.

Key Concepts

Probability of Compound Events

This concept involves calculating the likelihood of multiple events occurring together. When selecting items without replacement, probabilities change after each selection, requiring careful consideration of dependent events.

Combinatorics and Counting Methods

Combinatorics helps count the number of ways to choose items from a set without regard to order. Using combinations (n choose k) allows us to find the total possible selections and favorable outcomes in problems like selecting cans from a pack.

Hypergeometric Distribution

This distribution models the probability of k successes in n draws from a finite population without replacement. It is ideal for problems where items are classified into two types, such as diet and regular soda cans, and selections affect subsequent probabilities.