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  1. 047Three stocks you cover report this quarter. Each beats consensus with probability 0.6, independently of the others. What is the chance that at least two of them beat?Probability and brainteasersCoreSell-side equity researchBuy-side equity research

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    0.648. At least two means exactly two or all three. Exactly two can happen three ways, each with chance 0.6 x 0.6 x 0.4 = 0.144, so 0.432 together. All three beat with 0.6 cubed, 0.216. Adding them gives 0.648. The complement checks it: none beat 0.064 plus exactly one 0.288 is 0.352, and 1 minus 0.352 is 0.648.

    How do you avoid missing an outcome?

    Think of three friends each deciding whether to come to dinner. At least two coming covers four different guest lists: each of the three possible pairs, and all three. List the outcomes by how many succeed, count the ways each count can happen, and multiply by the chance of one such way. With three stocks there are only eight results, so write them down; with more, the count of ways is a binomial coefficientThe number of ways to choose k items out of n, written n choose k; here 3 choose 2 is 3..

    All eight results for three stocks, grouped by how many beat (B) or miss (M)0.216BBB0.144BBM0.144BMB0.144MBB0.096BMM0.096MBM0.096MMB0.064MMM3 beats: 0.2162 beats: 0.4321 beat: 0.2880 beats: 0.064At least two beatAdd the groupThree: 0.216Two: 3 x 0.144= 0.4320.648Not 0.6 x 0.6= 0.36
    Of the eight possible results, all three beating has chance 0.216 and each of the three ways to get exactly two beats has 0.144, so the at-least-two group adds to 0.648 rather than 0.6 x 0.6.
    The relationship
    P(X≥2)=(32)(0.6)2(0.4)+(0.6)3=0.432+0.216=0.648P(X \ge 2) = \binom{3}{2}(0.6)^2(0.4) + (0.6)^3 = 0.432 + 0.216 = 0.648
    Xthe number of the three stocks that beat
    3 choose 2the three ways to pick which two beat
    0.4the chance the remaining stock misses
    What it says in wordsAdd the chance of exactly two beats, counted three ways, to the chance of all three.

    Is independence a fair assumption for three stocks you cover?

    Usually not, and saying so is the part that sounds like an analyst. Stocks in one sector share demand, input costs and the same consensus-setting habits, so beats tend to cluster: when one beats, the others are more likely to. Clustering fattens both ends. The chance of all three beating and of none beating both rise above 0.216 and 0.064, and whether at least two beat can move either way, so the independent answer is a starting point, not a forecast.

    Where candidates lose it

    The fast wrong answer is 0.36, multiplying two 0.6s. It is the chance that two named stocks both beat, and it ignores both the choice of pair and the third stock.

    The second loss is getting exactly two, 0.432, and stopping. At least two includes all three. Use the complement as a check: it takes five seconds and catches both slips.

    What the interviewer asks next

    • What is the chance that exactly one of the three beats?
    • With ten stocks at 0.6 each, what is the expected number of beats?
    • If beats were perfectly correlated, what would the chance of at least two be?