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  1. 065A banker is running five independent live deals, and each has a 30% chance of closing this quarter. What is the probability that at least one closes?ProbabilityWarm upBulge bracket IBSales and trading

    Try it first

    Pick the probability that at least one deal closes.

    Show the worked solution

    About 83.2%. Go through the back door. Each deal fails to close with probability 0.7, and because the deals are independent the chance that all five fail is 0.7 to the fifth power, about 16.8%. At least one closing is the only other possibility, so it is 1 minus 16.8%, which is 83.2%. Adding five lots of 30% to get 150% is the trap.

    Why can you not add the five chances?

    If five friends each have a 30% chance of turning up to dinner, you would not say there is a 150% chance someone comes. Adding counts the evenings when two or three friends arrive several times over. Probabilities of separate events add only when the events cannot both happen, and here two deals can close in the same quarter. Five times 30% is instead the expected number of deals that close, 1.5, which is a useful number but a different question.

    Chances of separate events do not add; chances of all of them failing multiplyAdding: 30% + 30% + 30% + 30% + 30%30%30%30%30%30%100% is the ceiling150%not a probabilityit double counts quarterswhere two or more closeChance that none has closed, deal by dealx 0.7 each timestart100%deal 170.0%deal 249.0%deal 334.3%deal 424.0%deal 516.8%At least one closes = 100% - 16.8%= 83.2%
    Stacking five 30% chances gives 150%, which cannot be a probability, while multiplying five independent 70% chances of failure gives 16.8% for no deal closing, so at least one closes with probability 83.2%.

    Why is the complement the fast route?

    At least one means one, or two, or three, or four, or five deals closing, and working each case out is slow. The opposite of at least one is none, and none is a single clean case: every deal fails. Independence lets you multiply the five 0.7s, and the number shrinks fast: 70%, 49%, 34.3%, 24.0%, then 16.8%. Subtract from 100% and you have the answer in one line. Whenever a question says at least one, reach for the complement before anything else.

    The relationship
    P(at least one)=1−(1−0.3)5=1−0.75=1−0.168=0.832P(\text{at least one}) = 1 - (1 - 0.3)^5 = 1 - 0.7^5 = 1 - 0.168 = 0.832
    0.3the chance one deal closes
    0.7the chance one deal does not close
    5the number of independent deals
    What it says in wordsThe chance that at least one closes is one minus the chance that every deal fails.

    What does the independence assumption hide?

    The question tells you the deals are independent, and you should say that you are leaning on it. In a real quarter the deals share a market: a rate shock or a credit squeeze that stalls one is likely to stall the others, so the true chance of at least one closing is lower than 83.2%. If you want to show range, add that the chance of exactly one closing is 5 x 0.3 x 0.7 to the fourth, about 36.0%, so most of the 83.2% is a single close rather than a flood.

    Where candidates lose it

    The fast wrong answer is 150%, and even candidates who know it is impossible sometimes try to rescue it by capping at 100%. The interviewer wants to hear the word complement within the first few seconds.

    The second miss is multiplying the 30%s instead of the 70%s, which gives the chance that all five close, about 0.24%. Say which event you are multiplying before you multiply.

    What the interviewer asks next

    • What is the probability that exactly two of the five close?
    • How many such deals would you need for a 95% chance of at least one closing?
    • If the five deals are all in one sector, how would you change your answer?