Conditional Probability in Gender and Birthday Context: An In-Depth Analysis

This article delves into a classic problem involving conditional probability and its implications when combining gender and specific events like a child's birthday. We will explore the given problem and provide detailed calculations to determine the probabilities after each question.

The Problem

Suppose you meet a parent who has 2 children. You ask her if she has at least one boy, and she answers, Yes. You then ask her if she has at least one boy born in a specific month (let's call it month M), and she answers, Yes again. The question is: What are the probabilities after each question that she has two boys?

Initial Probabilistic Considerations

The wording of the question implies that we are trying to make an estimation of a probability based on the information available. Our initial estimates can change with new information, but the actual situation stays the same. This is a fundamental concept in probability theory. Let's denote the following events:

nA She has two boys. nB She has at least one boy with his birthday due next month (month M).

Without further knowledge, the probability of nA is:

[ PA left(frac{1}{2}right)^2 frac{1}{4} ]

The probability of nB is:

[ PB 1 - left(frac{1}{2}right)^2 frac{3}{4} times frac{11}{12} frac{33}{48} frac{33}{48} frac{47}{576} ]

Calculating Conditional Probability

We aim to find the probability of nA occurring given that nB is true. Using Bayes' theorem, we have:

[ P_{AB} frac{P_B A}{P_B} ]

First, we need to calculate PBA which is the probability that at least one child is a boy with a birthday in month M given that she has two boys:

[ P_{BA} frac{11 times 11}{12 times 12} frac{121}{144} ]

Now, we can calculate:

[ P_{AB} frac{121/144}{47/576} frac{121 times 576}{144 times 47} frac{121 times 4}{47} frac{484}{47} frac{23}{47} ]

Final Probabilities

After the first question, the probability she has two boys is:

[ P_{BA} frac{1}{3} ]

After the second question, the probability she has two boys is:

[ P_{AB} frac{23}{47} ]

Conclusion

The probability that the parent has two boys does change after the second question based on new information. The date of the birthday is indeed relevant in this context. Without further details, the initial probability is 1/4, but after confirming a boy's birthday, the probability increases to 23/47. This problem highlights the importance of conditional probabilities in making informed estimates based on available information.