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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.
Define the Events:
Identify Possible Outcomes:
Since at least one child is named Tom, we can eliminate the GG scenario. Thus, the remaining possibilities are:
Calculate Prior Probabilities:
Therefore,
Apply Bayes' Theorem:
Bayes' Theorem states:
P(A∣B)=P(B)P(B∣A)×P(A)
Substituting the values:
P(A∣B)=0.43750.75×0.25=0.43750.1875≈0.4286
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.