Hedge Funds puzzles, solved step by step

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  1. 052A family has two children and you learn that at least one of them is a girl. What is the probability that both are girls? How does the answer change if you learn instead that the elder child is a girl?Conditional probability and BayesWarm upQuant and systematic fundsProp and quant trading firms

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    At least one child is a girl. What is the chance both are?

    Show the worked solution

    One in three if you learn at least one is a girl, and one in two if you learn the elder is a girl. List the four equally likely families by birth order: boy-boy, boy-girl, girl-boy, girl-girl. At least one girl rules out only boy-boy and leaves three, one of which is girl-girl. Naming the elder rules out two orders and leaves girl-boy and girl-girl.

    Why is the answer not simply one half?

    Picture a friend tossing two coins behind a screen and telling you that at least one came up heads. She has told you something about the pair, not about a particular coin, so you cannot treat the other coin as a fresh toss. The pair had four equally likely outcomes and her remark removes only one of them, tails-tails. Two children work the same way, provided each birth is equally likely to be a boy or a girl and the two births are independent.

    Strike the cells each statement rules out, then count what is leftYou learn: at least one is a girlYounger boyYounger girlElder boyB-BB-GElder girlG-BG-GG-G is 1 of 3 cells left: 1/3You learn: the elder is a girlYounger boyYounger girlElder boyB-BB-GElder girlG-BG-GG-G is 1 of 2 cells left: 1/2
    Of the four equally likely birth orders, at least one girl strikes out only boy-boy and leaves three cells, so both girls is 1 in 3, while the elder is a girl keeps only the two cells of the elder-girl row, so both girls is 1 in 2.
    The relationship
    P(GG∣at least one G)=P(GG)P(at least one G)=1/43/4=13P(GG \mid \text{at least one } G) = \frac{P(GG)}{P(\text{at least one } G)} = \frac{1/4}{3/4} = \frac{1}{3}
    GGboth children are girls
    1/4the chance of girl-girl among four equally likely orders
    3/4the chance of at least one girl: every order except boy-boy
    What it says in wordsA conditional probability is the chance of both things happening divided by the chance of the thing you were told.

    What changes when you learn that the elder is a girl?

    Now the information is about a named child. Naming the child removes a whole row of the grid, both orders in which the elder is a boy, so two cells survive and the answer becomes one half. The younger child's sex is untouched by what you learned, which is why it now behaves like a fresh toss. The same arithmetic, 1/4 divided by 1/2, gives 1/2.

    Why would a fund interviewer ask this?

    Because every piece of market news arrives through a filter, and the filter changes what the news means. How you came to learn a fact is part of the fact. A screen that says at least one of two stocks beat estimates tells you less about each stock than a screen that names the one that did. Draw the grid, give both answers, and say the assumption out loud: independent births, each equally likely to be a boy or a girl.

    Where candidates lose it

    Nearly everyone answers one half for both versions, because the second child feels like a separate coin toss. The interviewer is checking whether you notice that at least one does not say which one.

    The other way to lose it is to reach one third and then say one third again for the elder-girl version. Draw the grid, strike the cells each statement rules out, and count what is left.

    What the interviewer asks next

    • You visit the family and a girl, chosen at random from the two children, opens the door. What is the chance both children are girls?
    • At least one child is a girl born on a Tuesday. What is the chance both are girls?
    • With three children and at least one girl, what is the chance all three are girls?