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Solution: The Boy-Girl Paradox – in a nutshell.

April 3, 2019

Solution to Exercise

a. The answer is 2/3. With two siblings, there are four possibilities of equal likelihood, i.e. sister – sister, brother – brother, sister – brother, brother –sister. Each element of the binary pairs can be distinguished by any unique discriminating factor, such as age or height or eye colour or location. We know that Cindy is her sister, so the brother-brother binary pair can be eliminated. This leaves the following possibilities. Sister – Sister (say older sister and younger sister); Sister – Brother (say older sister and younger brother); Brother – Sister (say older brother and younger sister). In two of these binary pairs, the other element is a Brother, whereas one binary pair contains a Sister. So the probability that her other sibling is a brother is 2/3. The problem is equivalent to learning that there are two coins, and being told that one of the two coins is heads up. Each element of the binary pairs can be distinguished by any unique discriminating factor, such as which hand the coin is held in. Now what is the probability that the other coin is also heads up? With two coins there are four possibilities. Heads – Heads, Heads – Tails, Tails – Heads, Tails – Tails. On being told that at least one of the coins is Heads, we can eliminate Tails – Tails. That leaves Heads – Heads (say left hand and right hand), Heads – Tails (say left hand and right hand) and Tails – Heads (say left hand and right hand). Of these three equally likely possibilities, two contain a Tails as the other element of the binary pair, one contains a Heads as the other element. So the probability that the other coin is a Tails is 2/3.

b. The answer is 1/2. With two children, there are four possibilities of equal likelihood, i.e. girl – girl, girl – boy, boy – girl, boy – boy. Each element of the binary pairs can be distinguished by any unique discriminating factor, such as age or height or eye colour or in this case location. We know that Barbara is her daughter, with her in the park, so the boy-boy binary pair can be eliminated. This leaves the following possibilities. Girl – Girl (girl in the park and girl at home); Girl – Boy (girl in the park and boy at home). We can eliminate Boy – Girl (boy in the park and girl at home). That leaves two binary pairs, Girl –Girl and Girl – Boy. In one of these binary pairs, the other element is a girl, whereas the other binary pair contains a boy. So the probability that her other sibling is a boy is 1/2. The problem is equivalent to learning that there are two coins, one in each hand and being told that one of the two coins (the one in the left hand is heads up). Each element of the binary pairs can be distinguished by any unique discriminating factor, and in this case it is the hand in which the coin is held. Now what is the probability that the other coin is also heads up? With two coins there are four possibilities. Heads – Heads, Heads – Tails, Tails – Heads, Tails – Tails. On being told that at least one of the coins is Heads, we can eliminate Tails – Tails. That leaves Heads – Heads (Heads in the left hand and Heads in the right hand), Heads – Tails (Heads in the left hand and Tails in the right hand). Tails – Heads (Tails in the left hand and Heads in the right hand can be eliminated as we know that Heads is in the left hand). Of these two equally likely possibilities, one contains a Tails as the other element of the binary pair and one contains a Heads as the other element. So the probability that the other coin is a Tails is 1/2.

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