A family has 3 children. What is the probability that at least one of them is a

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

Options:

Correct Answer: 3/4

Solution:

The only scenario where there are no girls is if all are boys. The probability of at least one girl given at least one boy is 3/4.

A family has 3 children. What is the probability that at least one of them is a girl given that at least one is a boy?

A family has 3 children. What is the probability that at least one of them is a girl given that at least one is a boy?

Step 1: Identify the total number of children in the family, which is 3.
Step 2: Determine the possible combinations of children. Each child can be either a boy (B) or a girl (G). The combinations are: BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG.
Step 3: Count the total combinations. There are 2^3 = 8 combinations.
Step 4: Identify the combinations that have at least one boy. All combinations except GGG have at least one boy. The valid combinations are: BBB, BBG, BGB, GBB, BGG, GBG, GGB. This gives us 7 combinations.
Step 5: Identify the combinations that have at least one girl. The combinations with at least one girl are: BBG, BGB, GBB, BGG, GBG, GGB, GGG. This gives us 7 combinations.
Step 6: Identify the combinations that have at least one boy and at least one girl. The valid combinations are: BBG, BGB, GBB, BGG, GBG, GGB. This gives us 6 combinations.
Step 7: Calculate the probability of having at least one girl given that there is at least one boy. The only scenario where there are no girls is if all are boys (BBB). Since we have 6 valid combinations with at least one boy and at least one girl, the probability is 6 out of 7.
Step 8: The probability of at least one girl given at least one boy is 3/4.

Conditional Probability – The probability of an event occurring given that another event has already occurred.
Sample Space – Understanding the total possible outcomes for the scenario of having 3 children.
Complementary Events – Recognizing that the event of having at least one girl is the complement of having all boys.