To determine the probability of a family with 3 children having exactly 1 boy and 2 girls, we assume the following:

1. Each child is equally likely to be a boy or a girl (probability = $\frac{1}{2}$).
2. The outcomes for each child are independent.

Step 1: Total Outcomes

Each child can be either a boy (B) or a girl (G). For 3 children, the total number of outcomes is: $2^3 = 8$ The possible outcomes are: $\{BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG\}$

Step 2: Favorable Outcomes

We are looking for outcomes with exactly 1 boy and 2 girls. These are: $\{BGG, GBG, GGB\}$ Thus, there are 3 favorable outcomes.

Step 3: Probability Calculation

The probability is given by: $\text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}$ $\text{Probability} = \frac{3}{8}$

Final Answer:

The probability that the family has exactly 1 boy and 2 girls is: $\boxed{\frac{3}{8}}$

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Would you like a detailed breakdown of how probabilities were assigned or further clarification? Here are 5 related questions to explore:

1. What is the probability of having exactly 2 boys and 1 girl in this scenario?
2. How would the probability change if the chance of having a boy were $60\%$ instead of $50\%$?
3. What is the probability of having at least one boy in a family with 3 children?
4. What is the probability of having no boys in a family with 3 children?
5. How many combinations are there for having exactly 2 girls and 1 boy in a family with 4 children?

Tip: When dealing with probabilities involving multiple events, consider using binomial coefficients to count favorable outcomes more efficiently!