Question

There are two children in a family. It is known that there is at least one child is a boy. Then find the probability that both children are boys.

Hint:

Be careful with conditional probability problems. The phrase "it is known that" or "given that" signals that you are no longer working with the original full sample space. You must restrict your analysis to the outcomes that satisfy the given condition.

Answer 1

Step 1: Understanding the Concept:
This is a conditional probability problem. We are given some information (the reduced sample space) and asked to find the probability of an event within that new sample space.
Step 2: Key Formula or Approach:
Let A be the event that both children are boys, and B be the event that at least one child is a boy. We want to find \( P(A|B) \), the probability of A given B.
The formula for conditional probability is: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] Step 3: Detailed Explanation or Calculation:
First, list the entire sample space S for two children (B=Boy, G=Girl): \[ S = \{BB, BG, GB, GG\} \] Each outcome has a probability of 1/4.
Now, define the events: - Event A: Both children are boys. \( A = \{BB\} \). The probability is \( P(A) = 1/4 \). - Event B: At least one child is a boy. This means we exclude the case where both are girls. \( B = \{BB, BG, GB\} \). The probability is \( P(B) = 3/4 \).
Next, we find the intersection of A and B, \( A \cap B \). This is the event where "both children are boys" AND "at least one child is a boy". This is simply the event "both children are boys". \[ A \cap B = \{BB\} \cap \{BB, BG, GB\} = \{BB\} \] The probability of the intersection is \( P(A \cap B) = 1/4 \).
Now, apply the conditional probability formula: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{1/4}{3/4} = \frac{1}{3} \] Alternative (Reduced Sample Space) Method:
The given information "at least one child is a boy" reduces our sample space to \( S' = \{BB, BG, GB\} \). All three outcomes in this new space are equally likely. We want to find the probability of the event "both children are boys" within this new sample space. The favorable outcome is \( \{BB\} \). \[ P(\text{both boys} | \text{at least one boy}) = \frac{\text{Number of favorable outcomes in } S'}{\text{Total number of outcomes in } S'} = \frac{1}{3} \] Step 4: Final Answer:
The probability that both children are boys is \( \frac{1}{3} \).