Throwing dice with an infinite number of sides

You have a die with an infinite number of equiprobable sides. Each side is assigned a value: 1,2,3,4… and so on

What are the chanses of you rolling ”a one” at least once if you try an infinite number of times?

The answer is that you have an approximate chance of 63.2% of doing that.

There is that number again, 63.2 !

The exact answer is

Some people may find that a trivial problem, but I think it’s interesting. I have never seen it expressed before.

That number is also at the root of my Hithchikers-Guide-Encoded-in-Pi problem and the Bonds Integer from my last post.


3 thoughts on “Throwing dice with an infinite number of sides

  1. Is ‘infinity’ an expression applicable in math, isn’t it just an idea? The thing I react to is the substitution of ((inf-1)/inf)^inf to 1/e.

  2. That ((x-1)/x)^x approaches the constant 1/e as x gets bigger is just a slight modification of one of the so called standard limits (standardgränsvärden). It can be proven in different ways.

    You can also empirically see what happens if you try out for example (9/10)^10 and then (99/100)^100 and then (999/1000)^1000 and then (9999/10000)^10000 and so on. If you continue like that you’ll get an ever increasingly precise calculation of 1/e.

    • Those proofs are not in my league of math yet, but I do have to agree that the substitution is correct mathematically. Both your empirical way and plotting the graph approaches 1/e.

      The logic however is another question… The quotient inside the brackets would go to a number very close to 1, but still not 1. Multiplying a number less than 1 with itself an infinite number of times would, in my sense, approach 0. The logic doesnt add upp.

      And then I believe it’s absurd that the chance of throwing a 1 at least once is not 100%. This makes me doubt whether math really is a great tool to describe the world.


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