Classic puzzle but I still second-guessed myself mid-answer.
Clarify the assumptions about how the information 'at least one is a boy' was obtained, then enumerate the equally likely outcomes consistent with that information. Compute the conditional probability by dividing the number of favorable outcomes by the total number of possible outcomes.
Pro tip: Explicitly state your assumptions about the selection process; this demonstrates that you understand the ambiguity and can avoid the common pitfall of assuming independence without conditioning.
Ask how it is known that at least one child is a boy. For example, was a random child identified as a boy, or was the family selected because it has at least one boy?
List all equally likely gender combinations for two children: BB, BG, GB, GG. Assume each is equally likely with probability 1/4.
Restrict the sample space to outcomes where at least one child is a boy: BB, BG, GB. This gives three equally likely outcomes.
The favorable outcome is BB. So the probability is 1 favorable outcome out of 3 possible outcomes, i.e., 1/3.
If the information was obtained by randomly selecting a child and observing they are a boy, the probability changes to 1/2. Mention this to show depth.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
This is where the variants start mattering.
Clarify that the problem is about conditional probability and that the older child being a boy is different from at least one child being a boy. Enumerate the sample space for two children ordered by age, then condition on the older child being a boy. Conclude that the probability both are boys is 1/2.
Pro tip: Explicitly contrast this with the classic 'at least one boy' variant, where the answer is 1/3, to show you understand how the information given changes the sample space.
List all equally likely outcomes for two children by birth order: BB, BG, GB, GG, where the first letter represents the older child.
The condition 'the older child is a boy' eliminates outcomes where the older child is a girl, leaving only BB and BG.
Among the remaining equally likely outcomes, only BB satisfies 'both are boys', so the probability is 1 out of 2, or 1/2.
Mention that if the condition were 'at least one child is a boy', the sample space would be BB, BG, GB, giving a probability of 1/3 for both boys.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
This one is genuinely weird and I love it in retrospect.
This is a classic conditional probability puzzle that tests careful reasoning about sample spaces. Start by clarifying the assumptions (e.g., independence, equal probability of boy/girl, and that 'born on a Tuesday' is a specific day). Then enumerate the equally likely outcomes for two children, filter those that satisfy the condition, and compute the ratio of favorable outcomes to total outcomes.
Pro tip: Don't rush to the intuitive answer (1/3 or 1/2); instead, explicitly state your assumptions and walk through the enumeration. This demonstrates rigorous thinking and avoids the common pitfall of misinterpreting the condition.
State that each child is equally likely to be a boy or girl, births are independent, and each day of the week is equally likely for a birth. Also clarify that 'at least one is a boy born on a Tuesday' means exactly one or both children meet this condition.
List all possible outcomes for two children, considering both gender and birth day. There are 2 genders × 7 days = 14 possibilities per child, so 14 × 14 = 196 equally likely outcomes for the pair.
Count the number of outcomes where at least one child is a boy born on a Tuesday. This includes cases where one child is a boy born on Tuesday and the other is anything, but avoid double-counting the case where both are boys born on Tuesday.
Among the outcomes satisfying the condition, count how many have both children as boys (regardless of birth day). This is the numerator for the probability.
Divide the number of favorable outcomes by the total number of outcomes satisfying the condition. Simplify the fraction to get the final probability.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Straightforward extension once you've done the two-child version.
Clarify that the problem assumes each child is equally likely to be a boy or girl and that the children are distinct. Then define the sample space of all 8 equally likely gender combinations, condition on the event that at least one is a boy, and compute the probability that all three are boys as the ratio of favorable outcomes to the conditioned sample space size.
Pro tip: Mention that the answer depends on how the information 'at least one is a boy' was obtained; if it came from randomly selecting a child and observing a boy, the probability changes. This shows you understand the subtlety of conditional probability and avoids ambiguity.
State that each child is independently a boy or girl with equal probability, and that the children are distinguishable (e.g., by birth order).
List all 8 equally likely outcomes: BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG.
The condition 'at least one is a boy' excludes only GGG, leaving 7 equally likely outcomes.
The event 'all three are boys' corresponds to the single outcome BBB.
The conditional probability is 1/7, since there is 1 favorable outcome out of 7 possible outcomes under the condition.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.