Hedge Funds puzzles, solved step by step
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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?Quant and systematic fundsProp and quant trading firms
Try it first
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.
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 relationshipGG both children are girls 1/4 the chance of girl-girl among four equally likely orders 3/4 the 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?
090Each of your analysts calls the direction of a stock correctly 70% of the time, independently of the others, and your prior is 50/50. Two analysts disagree. What is your probability now that the stock goes up? What if a third analyst then sides with the one who said up?Quant and systematic fundsProp and quant trading firms
Try it first
Where are you after the disagreement, and after the third call?
Show the worked solution
50% after the disagreement, and 70% once the third analyst sides with up. Work in odds. Each analyst's call multiplies the odds by 0.7/0.3 = 7/3 in the direction called. One up and one down multiply by 7/3 and 3/7, which cancel, leaving the prior of 1:1. The third call multiplies by 7/3 again: odds of 7:3, a probability of 70%. A two-to-one split is worth one analyst, not two thirds.
Why switch from probabilities to odds?
Think of two friends who read the weather equally well and disagree about rain: you are back where you started, however good they are. With independent signals, each one multiplies your odds by its likelihood ratioHow much more likely a piece of evidence is if the claim is true than if it is false., so in odds form Bayes' rule is just multiplication. An analyst who is right 70% of the time says up with chance 0.7 if the stock will rise and 0.3 if it will fall, so an up call multiplies the odds by 7/3 and a down call by 3/7.
Starting from 50%, analyst A's up call moves the chance of up to 70%, analyst B's down call returns it to 50%, and analyst C's up call moves it to 70% again, not to the 67% a vote count suggests; three up calls and none down would give 92.7%. How do the three calls combine?
Start at 1:1. The first analyst says up: 7:3, or 70%. The second says down: 7:3 times 3:7 is 1:1, back to 50%. The third says up: 7:3 again, 70%. Only the net count of calls matters, not the total. Three for up and none against would give 343:27, about 92.7%, which shows how much the one dissenter costs.
The relationship1 the prior odds of up against down, 50/50 7/3 the likelihood ratio of an up call from a 70% accurate analyst 3/7 the likelihood ratio of a down call What it says in wordsMultiply the prior odds by one likelihood ratio per independent call, then turn the odds back into a probability.What assumption carries all of this?
Independence. If the analysts read the same research and speak to the same management teams, their errors are correlated, and a second agreeing call adds much less than a factor of 7/3. In the extreme where the second analyst simply copies the first, it adds nothing at all. The rule also assumes each analyst is right 70% of the time whichever way the stock moves; an analyst who calls up too often tells you more when calling down. State independence as the assumption, then say how you would test it: by checking how often the analysts' past calls agreed with each other.
Where candidates lose it
The vote-counting answer, two out of three so 67%, is the usual loss. It treats a majority as a probability, when a two-to-one split carries exactly one net call of evidence, 70%.
The second loss comes after the disagreement: candidates reach for something like 58%, feeling that two good analysts must add something. With equal accuracy and opposite calls, the evidence cancels exactly.
What the interviewer asks next
- One analyst is right 80% of the time and the other 60%. The better one says up, the other down. Where are you?
- Five analysts split three to two. What is your probability?
- How would you estimate how correlated your analysts' calls are, and how would you adjust for it?