Does the king have a sister?

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Offline Saba Fatema

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Does the king have a sister?
« on: June 24, 2012, 04:17:59 PM »
Consider the following problem to test our skills with conditional probability:

The king comes from a family of two children. What is the probability that the other child is his sister?

The sample space for this problem can be considered to be the set S of four pairs (B, B), (B, G), (G, B), (G, G), where B stands for "boy" and G stands for "girl" and the first and second positions in the pair denote first and second born children, respectively. To be able to do the problem some assumptions must be made. Once again, we shall assume each of the four outcomes is equally likely.
 
This problem is tricky-a lot of people think the answer should be 1/2. If the question had been "what is the probability that a person's sibling is a sister," then the answer would be 1/2. But in the given problem you are sneakily given the information in the wording of the problem that one child, the king, is male, and that information eliminates the outcome (G, G) in the sample space as a possibility. The remaining three outcomes of S become the conditional, or updated, sample space S' of which two outcomes have a B and a G, (B, B), (B, G), (G, B).
Therefore, the probability that the other child is his sister is 2/3.

Further reading
Isaac, R. The Pleasures of Probability, Undergraduate texts in Mathematics, Springer, 1995.
« Last Edit: June 24, 2012, 04:26:19 PM by Saba Fatema »
Saba Fatema
Senior Lecturer
Department of GED
FSIT, DIU