Kim has 5 pairs of shoes; each pair is a different color. If Kim randomly selects 2 shoes without replacement...

1. Translate the problem requirements: Kim has 10 total shoes (5 pairs, each pair a different color). She selects 2 shoes randomly without replacement. We need the probability that both shoes are the same color (i.e., from the same pair).
2. Count total possible outcomes: Determine how many ways Kim can select 2 shoes from 10 total shoes.
3. Count favorable outcomes: Determine how many ways Kim can select 2 shoes that form a matching pair (same color).
4. Calculate probability: Use the ratio of favorable outcomes to total outcomes to find the probability.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we have here in simple terms. Kim owns 5 pairs of shoes, and each pair is a different color - maybe she has red shoes, blue shoes, green shoes, black shoes, and white shoes. Since each "pair" means 2 shoes of the same color, she has 10 individual shoes total.

Now Kim is going to randomly pick 2 shoes from her collection without looking. The question asks: what's the chance that the 2 shoes she picks will be the same color (meaning they form a complete pair)?

Process Skill: TRANSLATE - Converting the problem language into a clear mathematical setup

2. Count total possible outcomes

To find the probability, we need to think about this step by step. First, let's figure out how many different ways Kim can select 2 shoes from her 10 total shoes.

Imagine Kim picks her first shoe - she has 10 choices. Then for her second shoe, since she's not replacing the first shoe, she has 9 remaining choices. So you might think there are \(10 \times 9 = 90\) ways.

But wait! This counts each pair of shoes twice. For example, picking the left red shoe first and then the right red shoe is the same as picking the right red shoe first and then the left red shoe. Since the order doesn't matter, we need to divide by 2.

Total ways to pick 2 shoes = \((10 \times 9) \div 2 = 90 \div 2 = 45\) ways

3. Count favorable outcomes

Now let's count how many ways Kim can pick 2 shoes that are the same color (a matching pair).

This is much simpler! Kim has exactly 5 pairs of shoes, and each pair represents one way to pick 2 shoes of the same color:

• She could pick both red shoes (1 way)
• She could pick both blue shoes (1 way)
• She could pick both green shoes (1 way)
• She could pick both black shoes (1 way)
• She could pick both white shoes (1 way)

So there are exactly 5 favorable outcomes.

Process Skill: VISUALIZE - Using concrete examples to count all the matching pairs

4. Calculate probability

Now we can find the probability using our basic probability formula:

Probability = Number of favorable outcomes ÷ Total number of possible outcomes
Probability = \(5 \div 45 = \frac{1}{9}\)

Let's verify this makes sense: Kim has a relatively small chance (1 out of 9) of randomly picking a matching pair, which seems reasonable since most of her shoe combinations would be mismatched.

4. Final Answer

The probability that Kim will select 2 shoes of the same color is \(\frac{1}{9}\).

This matches answer choice C: \(\frac{1}{9}\).

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "without replacement" as "with replacement"

Students may overlook the crucial phrase "without replacement" and assume Kim puts the first shoe back before selecting the second. This would lead to calculating total outcomes as \(10 \times 10 = 100\) instead of the correct \(10 \times 9 = 90\) (before dividing by 2). This fundamentally changes the denominator and leads to an incorrect probability calculation.

2. Confusing the favorable outcome definition

Students might misinterpret "2 shoes of the same color" and think they need to count individual shoes rather than complete pairs. They may incorrectly think there are 10 favorable outcomes (counting each individual shoe) instead of recognizing that there are exactly 5 ways to select matching pairs.

Errors while executing the approach

1. Forgetting to divide by 2 when calculating total combinations

Even when students correctly identify that Kim has \(10 \times 9\) ways to select shoes without replacement, they often forget that order doesn't matter in combinations. They may use 90 as the total number of outcomes instead of the correct 45, leading to a probability of \(\frac{5}{90} = \frac{1}{18}\) instead of \(\frac{1}{9}\).

2. Arithmetic errors when simplifying the fraction

Students may correctly calculate \(\frac{5}{45}\) but make errors when simplifying this fraction. Common mistakes include incorrectly reducing it to \(\frac{1}{10}\) or \(\frac{1}{5}\) instead of the correct \(\frac{1}{9}\), often due to careless division or not finding the greatest common divisor properly.

Errors while selecting the answer

1. Selecting the unsimplified fraction equivalent

If students calculate the correct probability as \(\frac{5}{45}\) but fail to simplify it to \(\frac{1}{9}\), they might look for \(\frac{5}{45}\) among the answer choices. Not finding it, they may incorrectly select a choice that seems "close" like \(\frac{1}{10}\) (choice D) rather than properly simplifying their answer first.