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  1. 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?Conditional probability and BayesHardQuant 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.

    Each call multiplies the odds: a disagreement cancels, a 2 to 1 split is one call50% up1 : 1Prior70% up7 : 3A says upx 7/350% up1 : 1B says downx 3/770% up7 : 3C says upx 7/3vote count: 67%(wrong)Three up, none down: odds 343 : 27, a 92.7% chance of up
    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 relationship
    oddspost=1×73×37×73=73⇒p=77+3=70%\text{odds}_{\text{post}} = 1 \times \tfrac{7}{3} \times \tfrac{3}{7} \times \tfrac{7}{3} = \tfrac{7}{3} \quad\Rightarrow\quad p = \frac{7}{7+3} = 70\%
    1the prior odds of up against down, 50/50
    7/3the likelihood ratio of an up call from a 70% accurate analyst
    3/7the 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?