I jumped to 1/2 immediately and had to backtrack.
Clarify the assumptions about how the information 'at least one is a boy' was obtained, then compute the probability under the standard assumption that all gender combinations are equally likely and the statement is a random observation. Present the answer as 1/3, but also discuss how different assumptions (e.g., a specific child is known to be a boy) lead to 1/2.
Pro tip: Explicitly state that the answer depends on the selection mechanism—this demonstrates the kind of rigorous thinking valued in data science and avoids the common pitfall of assuming the answer is always 1/2.
Ask or state how the information 'at least one is a boy' was obtained. Assume that the family was randomly selected from all two-child families and that you are told at least one is a boy, with no other information.
List all equally likely gender combinations for two children: BB, BG, GB, GG. Since at least one is a boy, eliminate GG, leaving BB, BG, GB.
Under the assumption, the remaining outcomes are equally likely. Only BB satisfies 'both are boys', so the probability is 1/3.
Explain that if instead you know a specific child (e.g., the older) is a boy, the probability is 1/2. This highlights the importance of the selection mechanism.
Connect to how assumptions and data collection methods affect probabilistic models and decision-making, emphasizing the need to question how data is generated.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
This one I got right but probably for the wrong reasons.
Clarify that the probability changes from 1/3 to 1/2 because specifying the older child is a boy eliminates the girl-boy and girl-girl outcomes, leaving only boy-boy and boy-girl. Then explain the key distinction: this is different from merely knowing at least one child is a boy, which leaves three equally likely outcomes.
Pro tip: Emphasize that the answer hinges on how the information is obtained—if you simply know 'at least one is a boy,' the probability is 1/3, but if you know the older child is a boy, it's 1/2. This shows you understand conditional probability beyond memorized puzzles.
List all equally likely gender combinations for two children: BB, BG, GB, GG, where the first letter represents the older child.
Restrict the sample space to outcomes where the older child is a boy: BB and BG.
Since both remaining outcomes are equally likely, the probability that both are boys is 1 out of 2, or 50%.
Explain that if you only knew at least one child is a boy, the sample space would be BB, BG, GB, giving a probability of 1/3. This highlights the importance of the specific information.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Clarify the assumptions about how the child was selected: if the family has two children and a randomly chosen child is a boy, the probability the other is a boy is 1/2. If instead we are told that at least one child is a boy, the probability both are boys is 1/3. State the assumption explicitly and compute accordingly.
Pro tip: Amazon interviewers value candidates who proactively state assumptions and avoid ambiguity. Mention the classic 'boy or girl paradox' and explain why the answer depends on the selection mechanism.
Ask whether the child was randomly selected from all children in the family, or whether the information 'at least one boy' was given without random selection. This distinction changes the probability.
For two children, the possible gender combinations are BB, BG, GB, GG, each with probability 1/4 under usual assumptions. If a child is randomly picked and is a boy, we condition on that event.
If a random child is a boy, the probability the other is a boy is 1/2. If we only know at least one is a boy, the probability both are boys is 1/3. Show the calculation using Bayes' theorem or counting.
Explain that the answer depends on the exact wording. The phrase 'a child is picked at random and happens to be a boy' implies random selection, so the answer is 1/2. If the problem meant 'at least one is a boy', the answer is 1/3.
State the final probability clearly, referencing the assumption made. For the given wording, the probability is 1/2.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.