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  1. 026A startup has a 25% chance of dying in any given year, independently of what happened the year before. What is the chance it is still alive after five years, and after ten?Probability and expected valueWarm upSeed and early-stage VCIndia VC

    Try it first

    Quick instinct: what is the chance the startup is still alive after five years?

    Show the worked solution

    About 23.7% after five years and about 5.6% after ten. Each year the company survives with probability 0.75, and the years are independent, so the chances multiply: 0.75 to the fifth is 0.237. Ten years is that number squared, 0.237 x 0.237, about 0.056. Half the companies are gone before the end of year three.

    Why do you multiply survival rather than add up the deaths?

    Think of a batch of 100 phones, each with a one in four chance of breaking in every year you own it. In year one about 25 break and 75 are left. In year two the 25% applies to those 75, not to the original 100, so about 19 break and 56 are left. A yearly death rate only acts on the companies still alive, so survival shrinks by the same factor each year instead of falling by the same amount. Adding 25% five times gives 125%, and a probability above 100% is the sign you are using the wrong operation.

    Share of startups still alive, 25% chance of dying each yearone in four100%Yr 075%Yr 156%Yr 242%Yr 332%Yr 423.7%Yr 518%Yr 613%Yr 710%Yr 88%Yr 95.6%Yr 10Each year keeps 0.75 of the survivorsYear 5: 0.75^5 = 0.237Year 10: 0.237 x 0.237 = 0.056Year ten is year five squared, because the same five-year factor applies twice
    With a 25% chance of dying every year, the share of startups alive falls to 23.7% at year five, below the one in four line, and to 5.6% at year ten, which is the five-year figure squared.
    The relationship
    S(n)=(1−d)nS(5)=0.755≈0.237S(10)=0.2372≈0.056S(n) = (1-d)^n \qquad S(5) = 0.75^5 \approx 0.237 \qquad S(10) = 0.237^2 \approx 0.056
    dthe chance of dying in any one year, 0.25
    nthe number of years
    S(n)the chance of still being alive after n years
    What it says in wordsSurvival after n years is the one-year survival chance multiplied by itself n times.

    How do you get 0.75 to the fifth in your head?

    Build it from squares. 0.75 squared is 0.5625, call it 0.56. Squared again, 0.56 x 0.56 is about 0.316, which is year four. One more 0.75 takes 0.316 to 0.237 for year five. Year ten is year five squared, so once you have 0.237 the second answer is one step: 0.237 x 0.237 is about 0.056. Saying the squaring route out loud shows the interviewer a method rather than a memorised number.

    What does a flat death rate mean for a seed portfolio?

    Two more numbers fall out of the same 25%. The average company lives four years, one over the yearly death rate, and half are gone within about 2.4 years, where 0.75 to the n crosses one half. A seed fund of 30 companies with this death rate expects only about 1.7 of them alive at year ten, which is why seed funds are sized for most companies failing. Say the limitation too: real death rates are not flat. They are highest in the first two years and fall for companies that find a market, so a flat 25% overstates late deaths and understates early ones.

    Where candidates lose it

    The fast wrong answer is zero, or some version of five times 25%, because the candidate adds the yearly chances. That treats a dead company as able to die again. The interviewer is checking whether you know that independent yearly chances multiply.

    The second loss is getting 23.7% and then working ten years from scratch, slowly and aloud. Square the five-year figure; it is quicker and it shows you see the structure.

    What the interviewer asks next

    • What yearly death rate leaves exactly half the companies alive after five years?
    • If the death rate is 40% in year one and 15% every year after, what is five-year survival?
    • A fund wants at least three companies alive at year ten. How many should it back at a 25% yearly death rate?