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We roll a six-symmetric dice until we get a roll of 6. What is the average number of throws (including the final roll of six) assuming all rolls are even numbers?

I do

$P_n\left(A|B\right)=\left(\frac{2}{6}\right)^{n-1}\cdot\frac{1}{6}=\frac{1}{2\cdot3^n}$,

$EN=\sum_{n=1}^\infty{\frac{n}{2\cdot3^n}}=\frac{1}{2}\cdot\frac{3}{4}=\frac{3}{8}$.

The answer given to this exercise is 1.5.

What's wrong?

Of course, the probability $P_n\left(A|B\right)$ above is not a right conditional probability. It should be like this:

$A$ — the event of receiving a sequence of twos and fours ending in a six,
$A_n$ — the event of receiving a sequence of twos and fours ending in a six of length $n$;

then

$\mathbb EN=\sum_{n=1}^\infty n\mathbb P\left(A_n\vert A\right)=\sum_{n=1}^\infty n\frac{\mathbb P\left(A_n\cap A\right)}{\mathbb P\left(A\right)}$,

$\mathbb P\left(A\right)=\sum_{n=1}^\infty \left(\frac{1}{6}\right)^{n-1}\cdot 2^{n-1}\cdot \frac{1}{6}=\frac{1}{4}$,

$\mathbb P\left(A_n\cap A\right)=\left(\frac{1}{6}\right)^{n-1}\cdot 2^{n-1}\cdot \frac{1}{6}=\frac{1}{2\cdot 3^n}$,

$\mathbb EN=1.5$.

Thank you.

RandomName
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    This "paradoxical", or at least unintuitive, computation is fairly well known. See this article for example. – lulu Jul 24 '23 at 17:14
  • Note: I don't understand your computation at all. Obviously it takes at least one throw, so the expected number must be $>1$. the usual "bad" computation is some variant of "there are three possible rolls here, $2,4,6$. Thus at any turn the chance that a good roll is a winner is $\frac 13$, so the expected number of trials is $3$." As I say, however, that simple argument is wrong for reasons the article I linked to describes in detail. – lulu Jul 24 '23 at 17:17

1 Answers1

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There is a 1/3rd chance to get 6.

In 3 throws you are likely to get a six.

Hence the average is a half of it, which is 3/2 = 1.5

A bit weird, but ok.

The problem in your calculation maybe that you are trying to find the throw where will you finitely get a 6 for sure and then half it, but getting a 6 in such manner in sure for infinite throws.

Aditya Gupta
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