A couple decides to have 4 children. If they succeed in having 4 children and each child is equally likely...

1. Translate the problem requirements: We need to find the probability of getting exactly 2 girls and 2 boys when having 4 children, where each child has a \(50\%\) chance of being either gender
2. Calculate total possible outcomes: Determine how many different gender combinations are possible for 4 children
3. Count favorable outcomes using arrangement logic: Find how many ways we can arrange exactly 2 boys and 2 girls among the 4 positions
4. Apply probability formula: Divide favorable outcomes by total outcomes to get the final probability

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what we're looking for in everyday terms. We have a couple who will have 4 children. Each time a child is born, it's like flipping a fair coin - there's a \(50\%\) chance of a boy and a \(50\%\) chance of a girl. We want to find the probability that out of these 4 children, exactly 2 will be girls and exactly 2 will be boys.

Think of it this way: we're looking for arrangements where we have the pattern of 2 G's (girls) and 2 B's (boys) in some order among the 4 positions.

Process Skill: TRANSLATE

2. Calculate total possible outcomes

Let's think about all the different ways 4 children can turn out in terms of gender. For each child, there are 2 possibilities (boy or girl). Since we have 4 children:

• First child: 2 possibilities
• Second child: 2 possibilities
• Third child: 2 possibilities
• Fourth child: 2 possibilities

Using the multiplication principle, the total number of possible gender combinations is:
\(2 \times 2 \times 2 \times 2 = 16\) total outcomes

We can verify this by listing some examples: BBBB, BBBG, BBGB, BBGG, BGBB, BGBG, BGGB, BGGG, GBBB, GBBG, GBGB, GBGG, GGBB, GGBG, GGGB, GGGG.

3. Count favorable outcomes using arrangement logic

Now let's count how many ways we can arrange exactly 2 boys and 2 girls among 4 positions. Think of this as: "In how many ways can we choose 2 positions out of 4 for the girls?" (The remaining 2 positions will automatically be for the boys).

Let's list them systematically by considering which 2 positions the girls occupy:

• Positions 1,2: GGBB
• Positions 1,3: GBGB
• Positions 1,4: GBBG
• Positions 2,3: BGGB
• Positions 2,4: BGBG
• Positions 3,4: BBGG

Counting these arrangements: 6 favorable outcomes

This makes sense because we're choosing 2 spots out of 4 for the girls, which can be done in 6 different ways.

Process Skill: VISUALIZE

4. Apply probability formula

Now we can calculate the probability using the basic probability formula:

Probability = Number of favorable outcomes / Total number of possible outcomes

Probability = \(\frac{6}{16} = \frac{3}{8}\)

Let's verify this makes sense: \(\frac{3}{8} = 0.375\), which means there's a \(37.5\%\) chance of getting exactly 2 boys and 2 girls. This seems reasonable - it's more likely than getting all boys or all girls, but not the most likely single outcome.

Final Answer

The probability that the couple will have exactly 2 girls and 2 boys is \(\frac{3}{8}\).

Looking at our answer choices, this matches option (A) \(\frac{3}{8}\).

Answer: (A) \(\frac{3}{8}\)

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "exactly 2 girls and 2 boys" as an ordering requirement: Students might think they need to find the probability of a specific sequence like GGBB, rather than understanding that any arrangement of 2 girls and 2 boys among the 4 children satisfies the condition. This leads them to calculate \(\frac{1}{16}\) instead of recognizing this as a combinations problem.

2. Confusing this with a conditional probability problem: Some students might overthink the problem and try to apply conditional probability formulas, thinking they need to account for dependencies between births, when in reality each birth is an independent event with probability \(\frac{1}{2}\) for each gender.

3. Failing to recognize this as a binomial probability scenario: Students might not see that this is a classic binomial probability problem (\(n=4\) trials, probability of success=\(\frac{1}{2}\), exactly \(k=2\) successes), leading them to use overly complicated approaches instead of the straightforward counting method.

Errors while executing the approach

1. Miscounting the favorable arrangements: When listing all possible arrangements of 2 girls and 2 boys, students often miss one or two arrangements or accidentally count duplicates. For example, they might list GGBB, GBGB, GBBG, BGGB, BGBG but forget BBGG, getting 5 instead of 6 favorable outcomes.

2. Calculation errors when computing total outcomes: Students might incorrectly calculate \(2^4\), getting 8 or 12 instead of 16, especially when working under time pressure. This leads to wrong denominators in their final probability fraction.

3. Arithmetic mistakes when simplifying the fraction: Even with the correct values (6 favorable outcomes out of 16 total), students might make errors when reducing \(\frac{6}{16}\) to \(\frac{3}{8}\), potentially getting \(\frac{6}{16} = \frac{1}{4}\) by incorrectly canceling numbers.

Errors while selecting the answer

1. Not simplifying the fraction to match answer choices: Students might arrive at the correct unsimplified answer \(\frac{6}{16}\) but fail to reduce it to \(\frac{3}{8}\), then not find \(\frac{6}{16}\) among the choices and either guess or incorrectly convert to a decimal and pick the closest option.

2. Selecting the complement probability: Students who correctly calculate that there are 6 favorable outcomes out of 16 might accidentally select \(\frac{10}{16} = \frac{5}{8}\) (the probability of NOT getting exactly 2 girls and 2 boys), though this specific value doesn't appear in the answer choices, leading to confusion and potentially picking the wrong option.