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Data Interview Question

Two Boys in a Family

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Solution & Explanation

To solve this problem, we need to determine the probability that a family with two children has two boys, given that one of the children is named Tom. This is a classic probability question that can be tackled using Bayes' Theorem and understanding of conditional probabilities.

Step-by-Step Solution

  1. Define the Events:

    • Let A be the event that the family has two boys (BB).
    • Let B be the event that at least one child is named Tom.
  2. Identify Possible Outcomes:

    • The possible combinations of children in the family, assuming equal likelihood, are:
      • BB (both boys)
      • BG (boy and girl)
      • GB (girl and boy)
      • GG (both girls)

    Since at least one child is named Tom, we can eliminate the GG scenario. Thus, the remaining possibilities are:

    • BB
    • BG
    • GB
  3. Calculate Prior Probabilities:

    • P(A) = Probability of having two boys = 1/4 (since there are four equally likely combinations: BB, BG, GB, GG)
    • P(B|A) = Probability that at least one child is named Tom given both are boys. Assuming each boy has a 50% chance of being named Tom (independence),
      • Probability that neither is Tom = (1 - 0.5) * (1 - 0.5) = 0.25
      • Probability that at least one is Tom = 1 - 0.25 = 0.75
    • P(B) = Probability that at least one child is named Tom:
      • For BB: 0.75 (as calculated above)
      • For BG: 0.5 (only the boy can be named Tom)
      • For GB: 0.5 (only the boy can be named Tom)

    Therefore,

    • P(B) = (0.75 * 1/4) + (0.5 * 1/4) + (0.5 * 1/4) = 0.75/4 + 0.5/4 + 0.5/4 = 0.4375
  4. Apply Bayes' Theorem:

    Bayes' Theorem states:

    P(AB)=P(BA)×P(A)P(B)P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}

    Substituting the values:

    P(AB)=0.75×0.250.4375=0.18750.43750.4286P(A|B) = \frac{0.75 \times 0.25}{0.4375} = \frac{0.1875}{0.4375} \approx 0.4286

Conclusion

The probability that both children are boys, given that one of them is named Tom, is approximately 0.4286 or 42.86%. This reflects the increased likelihood due to the condition of having at least one boy named Tom, which affects the distribution of probabilities among the possible scenarios.