Probability Introduction for Children

This question asks us to find a specific probability related to the gender of two children in a family. It's a classic example of a conditional probability problem, where we are given some information (the older child is a boy) and asked to calculate a probability based on that new, reduced set of possibilities.

Understanding Two Children Scenarios

When a couple has two children, there are several possible combinations for their genders. Assuming that each child has an equal chance of being a boy (B) or a girl (G), and that the genders of the two children are independent events, we can list all possible outcomes. We consider the order of birth, so the older child is listed first, and the younger child second.

From the table, we can see there are 4 equally likely possible outcomes when a couple has two children.

Conditional Probability Application

The question provides a crucial piece of information: "if the older one is a boy". This is a condition that narrows down our possible outcomes. We are no longer looking at all 4 possibilities, but only those where the older child is a boy.

• From our initial list (BB, BG, GB, GG), we need to select only those where the first child (older one) is a boy.
• These outcomes are: BB and BG.

So, our new, reduced sample space, given the condition that the older child is a boy, consists of 2 outcomes:

1. BB (Older child is a boy, Younger child is a boy)
2. BG (Older child is a boy, Younger child is a girl)

Both of these outcomes are equally likely within this reduced sample space.

Calculating Both Children Probability

Now, we need to find the probability that "both children are boys" from this reduced sample space (BB, BG).

• The event "both children are boys" corresponds to the outcome BB.
• Within our reduced sample space (BB, BG), there is only 1 outcome where both children are boys.
• The total number of outcomes in our reduced sample space is 2.

The formula for probability is:

\[ \text{P(Event)} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]

In this case:

• Favorable outcomes (both children are boys, given older is a boy): 1 (BB)
• Total possible outcomes (older child is a boy): 2 (BB, BG)

Therefore, the probability that both children are boys if the older one is a boy is:

\[ \text{P(Both Boys | Older is Boy)} = \frac{1}{2} \]

Probability Conclusion

The probability that both children are boys, given that the older one is a boy, is \(\frac{1}{2}\). This makes sense because once we know the older child is a boy, the gender of the younger child is the only remaining unknown variable affecting whether both are boys. The younger child has a 1 in 2 chance of being a boy.