On average, how many times must I roll a dice until I get a $6$?

On average, how many times must I roll a dice until I get a $6$?

I got this question from a book called Fifty Challenging Problems in Probability.

The answer is $6$, and I understand the solution the book has given me. However, I want to know why the following logic does not work: The chance that we do not get a $6$ is $5/6$. In order to find the number of dice rolls needed, I want the probability of there being a $6$ in $n$ rolls being $1/2$ in order to find the average. So I solve the equation $(5/6)^n=1/2$, which gives me $n=3.8$-ish. That number makes sense to me intuitively, where the number $6$ does not make sense intuitively. I feel like on average, I would need to roll about $3$-$4$ times to get a $6$. Sometimes, I will have to roll less than $3$-$4$ times, and sometimes I will have to roll more than $3$-$4$ times.

Please note that I am not asking how to solve this question, but what is wrong with my logic above.

Thank you!

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You can calculate the average this way also.

The probability of rolling your first $6$ on the $n$-th roll is $$\left[1-\left(\frac{5}{6}\right)^n\right]-\left[1-\left(\frac{5}{6}\right)^{n-1}\right]=\left(\frac{5}{6}\right)^{n-1}-\left(\frac{5}{6}\right)^{n}$$

So the weighted average on the number of rolls would be
$$\sum_{n=1}^\infty \left(n\left[\left(\frac{5}{6}\right)^{n-1}-\left(\frac{5}{6}\right)^{n}\right]\right)=6$$

Again, as noted already, the difference between mean and median comes in to play. The distribution has a long tail way out right pulling the mean to $6$.
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For those asking about this graph, it is the expression above, without the Summation. It is not cumulative. (The cumulative graph would level off at $y=6$). This graph is just $y=x\left[\left(\frac{5}{6}\right)^{x-1}-(\left(\frac{5}{6}\right)^{x}\right]$

It’s not a great graph, honestly, as it is kind of abstract in what it represents. But let’s take $x=4$ as an example. There is about a $0.0965$ chance of getting the first roll of a $6$ on the $4$th roll. And since we’re after a weighted average, that is multiplied by $4$ to get the value at $x=4$. It doesn’t mean much except to illustrate why the mean number of throws to get the first $6$ is higher than around $3$ or $4.$

You can imagine an experiment with $100$ trials. About $17$ times it will only take $1$ throw($17$ throws). About $14$ times it will take $2$ throws ($28$ throws). About $11$ times it will take $3$ throws($33$ throws). About $9$ times it will take $4$ throws($36$ throws) etc. Then you would add up ALL of those throws and divide by $100$ and get $\approx 6.$

Your calculation is almost correct, but it’s calculating the wrong thing.

${(5/6)}^{n-1}$ is the probability that you roll any other number at least $n$ times until rolling a six. Setting this to $1/2$ gives:

$$n = \frac{-1}{\log_2 (5/6)}+1$$

This is the median of the distribution: the numerical value separating the higher half of the distribution from the lower half.

It is not the mean (average).

The experiment is: roll a die until you get a six.

The median is $3.8:$ That means that half the time when you perform this experiment you will get your six in under $3.8$ rolls and half the time you won’t.

The expected value is $6.$ This means that if you performed the experiment a hundred times and added all the rolls from each experiment together you should get around $600$ total rolls. So one could get the same total by assuming we had $6$ rolls in each experiment.

Think of it like this: although you have a $50\%$ chance of it taking less than $3.8$ rolls there are still gonna be a lot of times where it takes $8, 9, 10$ or more. Those high numbers are going to skew your expected values and leave you with an average of $6.$

The question you asked is a good one and goes right to the heart of what we mean by expected value.

The probability of the time of first success is given by the Geometric distribution.

The distribution formula is:

$$P(X=k) = pq^{n-1}$$

where $q=1-p$.

It’s very simple to explain this formula. Let’s assume that we consider as a success getting a 6 rolling a dice. Then the probability of getting a success at the first try is

$$P(X=1) = p = pq^0= \frac{1}{6}$$

To get a success at the second try, we have to fail once and then get our 6:

$$P(X=2)=qp=pq^1=\frac{1}{6}\frac{5}{6}$$

and so on.

The expected value of this distribution answers this question: how many tries do I have to wait before getting my first success, as an average? The expected value for the Geometric distribution is:

$$E(X)=\displaystyle\sum^\infty_{n=1}npq^{n-1}=\frac{1}{p}$$

or, in our example, $6$.

Edit: We are assuming multiple independent tries with the same probability, obviously.

The probability of something happening in n rolls might be 1/2, and that number might be say ’10’ – what if there was then a situation where the probability of the same event happening between 1000 and 2000 times was 1/2 – so 1-10 is P=1/2 1000-2000 is 1/2

The above could all make sense, but you can see that the average is never going to be 10 only.

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After your first roll, you either get a 6 and finish in 1 (probability 1/6), or you get a non-six and are back in the same position you were in at the start, with an expectation of a further E rolls needed (plus the one you made) – probability (5/6)

E = 1/6 + 5/6(E + 1)

(1/6)E = 1

E = 6