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  1. 011Diligence flags two independent risks in a target: a 20% chance its revenue will need to be restated, and a 10% chance its largest customer leaves. What is the chance at least one of them happens?Probability and expected value in dealsWarm upMid-market buyout fund

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    28%. The easy route is through the opposite event. The chance there is no restatement is 80% and the chance the customer stays is 90%; because the risks are independent, the chance of neither is 0.8 x 0.9, which is 72%. At least one problem is everything else, 100% less 72%, or 28%. Adding 20% and 10% gives 30% and double counts the 2% chance of both.

    Why not just add the two chances?

    Picture two friends who each might be late for dinner. If you add their chances of being late, the evening where both are late gets counted once for each of them. Adding probabilities works only when the events cannot happen together; when they can, the overlap is counted twice. Here both problems happen together 0.2 x 0.1, or 2%, of the time, so 20 plus 10 overshoots by exactly that: 30 less 2 is 28.

    Every outcome in one square: at least one problem is everything except the big cell2%customer only 8%restateonly18%neither problem0.8 x 0.9 = 72%restated 20%not restated 80%customer leaves 10%customer stays 90%At least one problem2 + 18 + 8 = 28%or 100 - 72 = 28%28%Adding 20 + 10 = 30%counts the red 2% celltwice, once in eachproblem's slice
    Splitting all outcomes 20 to 80 for the restatement and 10 to 90 for the customer gives four cells, 2%, 18%, 8% and 72%, so at least one problem is 28%, and adding 20% and 10% counts the 2% overlap twice.

    Why is one minus none the safest route?

    Because the opposite of at least one is a single clean case: nothing goes wrong. At least one problem is everything except the case where neither happens, so you multiply the two chances of no problem and subtract from one. It scales without effort. With five independent risks of 10% each, the chance none happens is 0.9 to the fifth, about 59%, so at least one is about 41%, a number that adding would put at 50%.

    The relationship
    P(at least one)=1−(1−0.2)(1−0.1)=1−0.72=0.28P(\text{at least one}) = 1 - (1 - 0.2)(1 - 0.1) = 1 - 0.72 = 0.28
    0.2chance of a revenue restatement
    0.1chance the largest customer leaves
    0.72chance neither happens, if the two are independent
    What it says in wordsThe chance of at least one problem is one less the chance of no problem at all.

    Is independence a fair assumption in diligence?

    Usually not, and saying so earns credit. A company that needs a revenue restatement may have weak controls or stretched customer relationships, so the two risks tend to move together, which raises the chance of both and lowers the chance of at least one below 28%. If the risks were perfectly linked, so the customer leaves only when revenue is also restated, the answer would fall to 20%. The 28% is the answer the question asks for; the correlation point is the judgement the interviewer is listening for.

    Where candidates lose it

    Saying 30% is the whole trap. It is quick, sounds right and is off by the overlap, which the interviewer chose small so that only careful candidates notice.

    The second miss is treating independence as a given. Answer 28%, then add one sentence on why diligence risks are rarely independent.

    What the interviewer asks next

    • What is the chance both happen?
    • With four independent 10% risks, what is the chance of at least one?
    • If the two risks are positively correlated, does the chance of at least one rise or fall?