Two Children, At Least One Boy
A family has two children. You are given the information that at least one of the children is a boy. What is the probability that both children are boys?
Related Concepts
Hint
- List all possible gender combinations for two children, assuming equal probability for Boy (B) and Girl (G) and independence between births. (e.g., BB, BG, GB, GG). This is your initial sample space.
- The information "at least one is a boy" reduces this sample space. Which outcome(s) from the initial list are now impossible given this new information?
- From this new, smaller sample space, identify the outcome(s) where "both are boys."
- The probability is the number of "both boys" outcomes divided by the total number of outcomes in the reduced sample space.
Be careful not to jump to an intuitive (but often incorrect) answer of 1/2!
Explanation: Two Children, At Least One Boy
Imagine a family has two children. We know for sure that at least one of them is a boy. What's the chance that the other child is also a boy (meaning they have two boys)?
This sounds tricky, and many people guess 1/2, but let's list the possibilities:
- All Possibilities for Two Kids (Order matters: older, younger):
- Boy, Boy (BB)
- Boy, Girl (BG)
- Girl, Boy (GB)
- Girl, Girl (GG)
- Apply the Given Information: "At least one is a boy." This means we can eliminate the (Girl, Girl) scenario, because that family has no boys.
Our new, smaller list of possibilities is:- Boy, Boy (BB)
- Boy, Girl (BG)
- Girl, Boy (GB)
- Find the Favorable Outcome: "Both are boys." Out of our new list of 3 possibilities, only one of them is (Boy, Boy).
- The Chance: So, there's 1 "both boys" case out of 3 possible cases (given that at least one is a boy). The probability is 1/3.
This is a classic conditional probability problem. Let B represent a boy and G represent a girl. We assume the probability of having a boy or a girl is equal (0.5) and the gender of each child is independent.
1. Define the Initial Sample Space
For two children, listing the older child first, the possible equally likely outcomes are:
- BB (Both Boys)
- BG (Older Boy, Younger Girl)
- GB (Older Girl, Younger Boy)
- GG (Both Girls)
There are 4 equally likely outcomes in the initial sample space.
2. Apply the Given Information (Condition)
We are given the information that "at least one child is a boy." This condition changes our effective sample space because it eliminates any outcome where this condition is not met.
Let E be the event "at least one child is a boy." The outcomes in E are:
- BB
- BG
- GB
The outcome GG (Both Girls) is excluded because it does not satisfy the condition of having at least one boy. So, our reduced sample space (or the event E which we know has occurred) now consists of these 3 equally likely outcomes.
3. Identify the Favorable Outcome within the Reduced Sample Space
We want to find the probability that "both children are boys" (event F), given that we know "at least one child is a boy" (event E).
Within our reduced sample space {BB, BG, GB}, the only outcome where both children are boys is:
- BB
There is 1 favorable outcome.
4. Calculate the Conditional Probability
The probability of event F (both boys) given event E (at least one boy) is:
P(F | E) = (Number of outcomes in F that are also in E) / (Number of outcomes in E)
Since F = {BB} and E = {BB, BG, GB}, the outcomes in F that are also in E is just {BB}.
P(Both Boys | At least one Boy) = 1 / 3
As a decimal, this is approximately:
1/3 ≈ 0.3333 or about 33.33%
Why not 1/2? A common mistake is to reason: "We know one is a boy, so we just need the other to be a boy, which is a 1/2 chance." This is incorrect because the information "at least one is a boy" is not equivalent to pointing to a specific child and saying "this child is a boy, what about the other?" The given information applies to the pair of children. The three scenarios (BB, BG, GB) are equally likely before we get any information, and after we know at least one is a boy, these three remain equally likely relative to each other.
Using Formal Conditional Probability Formula: Let F be the event {BB}. Let E be the event {BB, BG, GB}. P(F) = 1/4 (from original sample space) P(E) = 3/4 (from original sample space) P(F ∩ E) is the probability of {BB} AND {BB, BG, GB}, which is just P({BB}) = 1/4. P(F|E) = P(F ∩ E) / P(E) = (1/4) / (3/4) = 1/3.
Variation to Consider: How does the probability change if the information given was: "The older child is a boy. What is the probability both are boys?" Why is this different?