[I actually wrote this in my personal notes years ago. Seemed like a good fit for quick takes.]
I just rediscovered something in math, and the way it came out to me felt really funny.
I was thinking about startup incubators, and thinking about how it can be worth it to make a bet on a company that you think has only a one in ten chance of success, especially if you can incubate, y’know, ten such companies.
And of course, you’re not guaranteed success if you incubate ten companies, in the same way that you can flip a coin twice and have it come up tails both times. The expected value is one, but the probability of at least one success is not one.
So what is it? More specifically, if you consider ten such 1-in-10 events, do you think you’re more or less likely to have at least one of them succeed? It’s not intuitively obvious which way that should go.
Well, if they’re independent events, then the probability of all of them failing is 0.9^10, or
(1−110)10≈0.35.
And therefore the probability of at least one succeeding is 1−0.35=0.65. More likely than not! That’s great. But not hugely more likely than not.
(As a side note, how many events do you need before you’re more likely than not to have one success? It turns out the answer is 7. At seven 1-in-10 events, the probability that at least one succeeds is 0.52, and at 6 events, it’s 0.47.)
So then I thought, it’s kind of weird that that’s not intuitive. Let’s see if I can make it intuitive by stretching the quantities way up and down — that’s a strategy that often works. Let’s say I have a 1-in-a-million event instead, and I do it a million times. Then what is the probability that I’ll have had at least one success? Is it basically 0 or basically 1?
...surprisingly, my intuition still wasn’t sure! I would think, it can’t be too close to 0, because we’ve rolled these dice so many times that surely they came up as a success once! But that intuition doesn’t work, because we’ve exactly calibrated the dice so that the number of rolls is the same as the unlikelihood of success. So it feels like the probability also can’t be too close to 1.
So then I just actually typed this into a calculator. It’s the same equation as before, but with a million instead of ten. I added more and more zeros, and then what I saw was that the number just converges to somewhere in the middle.
1−(1−11000000)1000000=0.632121
If it was the 1300s then this would have felt like some kind of discovery. But by this point, I had realized what I was doing, and felt pretty silly. Let’s drop the “1−”, and look at this limit;
limn→∞(1−1n)n
If this rings any bells, then it may be because you’ve seen this limit before;
e=limn→∞(1+1n)n
or perhaps as
ex=limn→∞(1+xn)n
The probability I was looking for was 1−1e, or about 0.632.
I think it’s really cool that my intuition somehow knew to be confused here! And to me this path of discovery was way more intuitive that just seeing the standard definition, or by wondering about functions that are their own derivatives. I also think it’s cool that this path made e pop out on its own, since I almost always think of e in the context of an exponential function, rather than as a constant. It also makes me wonder if 1/e is more fundamental than e. (Similar to how 2π is more fundamental than π.)
There’s a piece of folklore whose source I forget, which says. “If someone in the hallway asks you a question about probability, then with probability 1/e the answer will be 1/e. The rest of the time it will be ‘you should switch.’”
Of course, at least in the context of startups, the success of the startups will be correlated, for multiple reasons, partly selection effects (selected by the same funders), partly network effects (if they are together in a batch, they will benefit (or harm) each other).
Rediscovering some math.
[I actually wrote this in my personal notes years ago. Seemed like a good fit for quick takes.]
I just rediscovered something in math, and the way it came out to me felt really funny.
I was thinking about startup incubators, and thinking about how it can be worth it to make a bet on a company that you think has only a one in ten chance of success, especially if you can incubate, y’know, ten such companies.
And of course, you’re not guaranteed success if you incubate ten companies, in the same way that you can flip a coin twice and have it come up tails both times. The expected value is one, but the probability of at least one success is not one.
So what is it? More specifically, if you consider ten such 1-in-10 events, do you think you’re more or less likely to have at least one of them succeed? It’s not intuitively obvious which way that should go.
Well, if they’re independent events, then the probability of all of them failing is 0.9^10, or
(1−110)10≈0.35.And therefore the probability of at least one succeeding is 1−0.35=0.65. More likely than not! That’s great. But not hugely more likely than not.
(As a side note, how many events do you need before you’re more likely than not to have one success? It turns out the answer is 7. At seven 1-in-10 events, the probability that at least one succeeds is 0.52, and at 6 events, it’s 0.47.)
So then I thought, it’s kind of weird that that’s not intuitive. Let’s see if I can make it intuitive by stretching the quantities way up and down — that’s a strategy that often works. Let’s say I have a 1-in-a-million event instead, and I do it a million times. Then what is the probability that I’ll have had at least one success? Is it basically 0 or basically 1?
...surprisingly, my intuition still wasn’t sure! I would think, it can’t be too close to 0, because we’ve rolled these dice so many times that surely they came up as a success once! But that intuition doesn’t work, because we’ve exactly calibrated the dice so that the number of rolls is the same as the unlikelihood of success. So it feels like the probability also can’t be too close to 1.
So then I just actually typed this into a calculator. It’s the same equation as before, but with a million instead of ten. I added more and more zeros, and then what I saw was that the number just converges to somewhere in the middle.
1−(1−11000000)1000000=0.632121If it was the 1300s then this would have felt like some kind of discovery. But by this point, I had realized what I was doing, and felt pretty silly. Let’s drop the “1−”, and look at this limit;
limn→∞(1−1n)nIf this rings any bells, then it may be because you’ve seen this limit before;
e=limn→∞(1+1n)nor perhaps as
ex=limn→∞(1+xn)nThe probability I was looking for was 1−1e, or about 0.632.
I think it’s really cool that my intuition somehow knew to be confused here! And to me this path of discovery was way more intuitive that just seeing the standard definition, or by wondering about functions that are their own derivatives. I also think it’s cool that this path made e pop out on its own, since I almost always think of e in the context of an exponential function, rather than as a constant. It also makes me wonder if 1/e is more fundamental than e. (Similar to how 2π is more fundamental than π.)
There’s a piece of folklore whose source I forget, which says. “If someone in the hallway asks you a question about probability, then with probability 1/e the answer will be 1/e. The rest of the time it will be ‘you should switch.’”
Oddly enough, you’re the second person to post about this.
Of course, at least in the context of startups, the success of the startups will be correlated, for multiple reasons, partly selection effects (selected by the same funders), partly network effects (if they are together in a batch, they will benefit (or harm) each other).
It’s the exponential map that’s more fundamental than either e or 1/e. Alon Amit’s essay is a nice pedagogical piece on this.