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A family has 3 children. What is the probability that at least one child is a gi

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Question: A family has 3 children. What is the probability that at least one child is a girl given that at least one child is a boy?

Options:

  1. 1/2
  2. 2/3
  3. 3/4
  4. 1/4

Correct Answer: 2/3

Solution: 

The only combinations with at least one boy are: BBB, BBG, BGB, GBB, BGG, GBG, GGB. Out of these, all combinations except BBB have at least one girl. Thus, P(At least one girl | At least one boy) = 6/7.

A family has 3 children. What is the probability that at least one child is a gi

Practice Questions

Q1
A family has 3 children. What is the probability that at least one child is a girl given that at least one child is a boy?
  1. 1/2
  2. 2/3
  3. 3/4
  4. 1/4

Questions & Step-by-Step Solutions

A family has 3 children. What is the probability that at least one child is a girl given that at least one child is a boy?
  • Step 1: Identify the total number of children in the family. There are 3 children.
  • Step 2: Determine the possible combinations of boys (B) and girls (G) for 3 children. The combinations are: BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG.
  • Step 3: Filter the combinations to find those that have at least one boy. The valid combinations are: BBB, BBG, BGB, GBB, BGG, GBG, GGB.
  • Step 4: Count the number of combinations that have at least one boy. There are 7 combinations: BBB, BBG, BGB, GBB, BGG, GBG, GGB.
  • Step 5: Now, count how many of these combinations have at least one girl. The combinations with at least one girl are: BBG, BGB, GBB, BGG, GBG, GGB. This gives us 6 combinations.
  • Step 6: Calculate the probability of having at least one girl given that there is at least one boy. This is done by dividing the number of combinations with at least one girl (6) by the total number of combinations with at least one boy (7).
  • Step 7: The probability is 6/7.
  • Conditional Probability – The question tests the understanding of conditional probability, specifically calculating the probability of an event given that another event has occurred.
  • Sample Space – It requires knowledge of how to construct a sample space for a scenario involving multiple children and their genders.
  • Combinatorial Analysis – The problem involves analyzing combinations of outcomes to determine the relevant probabilities.
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