Probability Calculation for Shoe Selection

This problem involves calculating the probability of a specific outcome when selecting a subset from a larger set. We are choosing 4 shoes from a closet containing 8 pairs (16 shoes total), and we want to find the probability that all 4 chosen shoes are of the same type (either all left or all right).

Understanding the Problem Setup

• Total number of shoe pairs: 8
• Total number of individual shoes: \(8 \text{ pairs} \times 2 \text{ shoes/pair} = 16\) shoes
• Number of left shoes: 8
• Number of right shoes: 8
• Number of shoes chosen at random: 4

Step 1: Calculate Total Possible Outcomes

The total number of ways to choose 4 shoes from the 16 available shoes is given by the combination formula \(C(n, k)\), where \(n\) is the total number of items to choose from, and \(k\) is the number of items to choose. In this case, \(n=16\) and \(k=4\).

Total outcomes = \(C(16, 4)\)

The formula for \(C(n, k)\) is \(\frac{n!}{k!(n-k)!}\).

Total outcomes = \(C(16, 4) = \frac{16!}{4!(16-4)!} = \frac{16!}{4!12!}\)

\(C(16, 4) = \frac{16 \times 15 \times 14 \times 13}{4 \times 3 \times 2 \times 1} = \frac{43680}{24} = 1820\)

There are 1820 total ways to choose 4 shoes from the 16 shoes in the closet.

Step 2: Calculate Favorable Outcomes

A favorable outcome is when all four chosen shoes are of the same type. This can happen in two mutually exclusive ways:

• All 4 shoes are left shoes.
• All 4 shoes are right shoes.

Let's calculate the number of ways for each case:

Case 1: All 4 shoes are left shoes

There are 8 left shoes. The number of ways to choose 4 left shoes from 8 is \(C(8, 4)\).

Number of ways to choose 4 left shoes = \(C(8, 4) = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!}\)

\(C(8, 4) = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = \frac{1680}{24} = 70\)

Case 2: All 4 shoes are right shoes

There are 8 right shoes. The number of ways to choose 4 right shoes from 8 is \(C(8, 4)\).

Number of ways to choose 4 right shoes = \(C(8, 4) = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} = 70\)

Total favorable outcomes = (Ways to choose 4 left shoes) + (Ways to choose 4 right shoes)

Total favorable outcomes = \(70 + 70 = 140\)

Step 3: Calculate the Probability

The probability of an event is calculated as:

Probability = \(\frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}\)

Probability = \(\frac{140}{1820}\)

Step 4: Simplify the Probability

We can simplify the fraction by dividing the numerator and the denominator by their greatest common divisor.

\(\frac{140}{1820} = \frac{14}{182}\) (dividing by 10)

Now, we can divide by 14 (since \(14 \times 13 = 182\)).

\(\frac{14 \div 14}{182 \div 14} = \frac{1}{13}\)

The probability that all four shoes chosen are of the same type is \(\frac{1}{13}\).

Revision Table: Key Concepts

Additional Information: Probability and Combinations

Probability questions often involve combinations or permutations to count the number of outcomes. When the order of selection does not matter, combinations are used, as in this shoe problem. If the question asked about the order in which the shoes were picked, we would use permutations.

In this problem, we calculated the total number of ways to pick any 4 shoes and the number of ways to pick 4 shoes specifically of the same type (all left or all right). By dividing the number of favorable outcomes by the total number of outcomes, we arrived at the required probability.

Understanding the difference between combinations and permutations is crucial for solving probability problems correctly. Combinations (\(C(n, k)\)) are used for selecting items (like choosing a hand of cards or selecting a group of shoes), while permutations (\(P(n, k)\)) are used for arranging items (like arranging letters in a word or finishing positions in a race).