Packaging Error
Because of a mistake in packaging, a case of 12 bottles of red wine contained five Merlot and seven Cabernet, each without labels. All the bottles look identical and any has an equal chance of being chosen. Three bottles are selected at random.
b. What is the probability that exactly two are Merlot?

Verified step by step guidance

1. Identify the total number of bottles and the composition: there are 12 bottles in total, with 5 Merlot and 7 Cabernet.
2. Determine the total number of ways to choose 3 bottles from the 12 without regard to order. This is given by the combination formula: \(\displaystyle \binom{12}{3}\).
3. Calculate the number of ways to choose exactly 2 Merlot bottles from the 5 available. This is \(\displaystyle \binom{5}{2}\).
4. Calculate the number of ways to choose the remaining 1 bottle as Cabernet from the 7 available. This is \(\displaystyle \binom{7}{1}\).
5. Use the multiplication rule to find the number of favorable outcomes by multiplying the two combinations from steps 3 and 4, then divide by the total number of ways from step 2 to get the probability: \(\displaystyle \frac{\binom{5}{2} \times \binom{7}{1}}{\binom{12}{3}}\).

Key Concepts

Hypergeometric Distribution

The hypergeometric distribution models the probability of drawing a specific number of successes from a finite population without replacement. In this problem, it helps calculate the chance of selecting exactly two Merlot bottles from the total of 12, given the fixed numbers of Merlot and Cabernet bottles.

Combinatorics (Combinations)

Combinations count the number of ways to choose items from a group without regard to order. Here, combinations are used to determine how many ways to select 2 Merlot bottles out of 5 and 1 Cabernet bottle out of 7, as well as the total ways to pick any 3 bottles from 12.

Probability Calculation

Probability is calculated as the ratio of favorable outcomes to total possible outcomes. For this question, the probability that exactly two bottles are Merlot equals the number of ways to choose 2 Merlot and 1 Cabernet divided by the total ways to choose any 3 bottles.