Consider the following experiment. I roll a die repeatedly until the die returns 6, then I count the number of times 3 appeared in the random variable $X$. What is $E[X]$?
Thoughts: I expect to roll the die 6 times before 6 appears (this part is geometric), and on the preceding 5 rolls each roll has a $1/5$ chance of returning a 3. Treating this as binomial, I therefore expect to count 3 once, so $E[X]=1$.
Problem: Don't know how to model this problem mathematically. Hints would be appreciated.
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Let $Y$ be the number of times the dice is rolled. Then $Y$ follows a geometric distribution with parameter $\frac16$.
We have $\mathbb{E}(Y) = 1 + 5 \mathbb{E}(X)$ since for the first $Y-1$ rolls each of the outcomes $1$, $2$, $3$, $4$, $5$ has equal probability.
It follows (using that a geometric distribution with parameter $p$ has mean $\frac{1}{p}$) that $$ 1 + 5 \mathbb{E}(X) = \mathbb{E}(Y) = 6 $$ so $\mathbb{E}(X) = 1$.
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Set $X_i$ to be the indicator random variable that you count a 3 on roll $i$, which in particular implies that you didn't stop before roll $i$. To do this, you need a non-six on $i-1$ rolls, and a three on the last roll. Hence $$E(X_i)=\left(\frac{5}{6}\right)^{i-1}\left(\frac{1}{6}\right)$$ $\sum X_i$ is the total number of threes you roll. By the linearity of expectation, $$E\left(\sum X_i\right)=\sum E(X_i)=\frac{1}{6}\sum_{i\ge 1} \left(\frac{5}{6}\right)^{i-1}=\frac{1}{6}\sum_{j\ge 0} \left(\frac{5}{6}\right)^{j}=\frac{1}{6}\frac{1}{1-\frac{5}{6}}=1$$
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You can apply the total expectation formula. This allows you to simplify the process of computing an expectation by summing over conditional expectations. There are lots of ways to choose how to do the conditioning; here is just one of them. Let $T$ be the number of $3$s counted in the whole process and let $N$ be the number of rolls in the process. Then
$$E[T]=\sum_{n=1}^\infty E[T\mid N=n] P[N=n].$$
The distribution of $N$ is geometric, while $E[T \mid N=n]$ is $n$ times the probability to get a $3$ on any given roll, i.e. $1/6$. So you're left to calculate
$$\sum_{n=1}^\infty \frac{n}{6} \left ( \frac{5}{6} \right )^{n-1} \frac{1}{6}.$$
Moving constants around, this is the same as
$$\frac{1}{36} \sum_{n=1}^\infty n \left ( \frac{5}{6} \right )^{n-1}.$$
This sum is pretty well-known; there is a way to approach it based on interchanging order of summation, and another way to approach based on differentiating the geometric series. The final answer should be $1$.
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There are infinite ways to solve this problem, here is another solution I like.
Let $A = \{\text{first roll is }6\}$, $B = \{\text{first roll is }3\}$, $C = \{\text{first roll is neither }3\text{ nor }6\}$. Then $$ E[X] = E[X|A]P(A) + E[X|B] P(B) + E[X|C] P(C) = 0 + (E[X] + 1) \frac16 + E[X]\frac46, $$ whence $E[X] = 1$.
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It turns out to be simple. Roll the die until you get a first $3$ or $6$. Then if you get a $6$, half the time, you stop with a value of $0$. Otherwise, you repeat, and the expected number of threes given the first three is $1+E(X)$. So the expected value of $X$ is:
$$E(X)=\frac{1}{2}\cdot 0 + \frac{1}{2}(1+E(X))= \frac{1}{2} + \frac{1}{2}E(X)$$
So $E(X)=1$.
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Let X be the number of throws until you get the first 6. It is easy to calculate that E(X)=6. Thus, if Y denotes the number of throws before the first 6, we have E(Y)=5. Therefore, for W denoting the number of 3's before the first 6 we have E(W)=1 as of these five throws every fifth on the average is a 3.
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Let's denote the expected number of 3's with $x$.
For the first throw, there are three possibilities:
So together we get: $$x = \frac16\cdot 0 + \frac16\cdot(x+1) + \frac46\cdot x$$ It is not hard to solve that equation for $x$, to get $$x = 1$$ Therefore the expected number of 3's is $1$.
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Let's break the problem down into two parts:
For the first question, if a $6$ comes up for the first time on the $n$th roll, then one of the other $5$ numbers came up on each of the $n-1$ previous rolls. On each of those rolls the probability of getting a $3$ is $1/5$, so we would expect that $3$ will occur $\frac{n-1}{5}$ times.
For the second question, the probability that it takes $n$ rolls to get to the first occurrence of a $6$ is given by $(5/6)^{n-1} \cdot (1/6)$.
Now to answer the question: You can think of the expected number of $3$s as the sum of the expressions that answer the first question above, weighted by the probabilities that those situations occur. That is, we need to calculate
$$\frac{0}{5} \cdot (5/6)^0 \cdot (1/6) + \frac{1}{5} \cdot (5/6)^1 \cdot (1/6) + \frac{2}{5} \cdot (5/6)^2 \cdot (1/6) + \frac{3}{5} \cdot (5/6)^3 \cdot (1/6) +\dots $$
We can factor this a bit:
$$\frac{1}{30} \left( \frac{5}{6} + 2\left( \frac{5}{6} \right)^2 +3\left( \frac{5}{6} \right)^3 + 4\left( \frac{5}{6} \right)^4 + \cdots\right) $$
The sum in the parentheses is of the form $\sum_{n=0}^{\infty}na^n $, which converges to $\frac{a}{(1-a)^2}$; in this case we have $a=5/6$ so the sum in parentheses is $\frac{5/6}{(1/6)^2} = 30$, and therefore the whole expression is just $1$.
In fact the naive intuition (it should take about $6$ rolls to get to a $6$, and on each of the previous $5$ rolls there is a $1$ in $5$ chance of getting a $3$, so it should happen once) leads you to the correct answer, but I am not sure if that is just good fortune or if it reflects some deeper truth. I suspect the latter.
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Imagine repeating the experiment a huge number of times, to obtain an estimate for the expected value. That estimate is simply the total number of $3$'s rolled divided by the total number of $6$'s rolled. In the limit the estimate equals the expected value. But in the limit, the ratio of $3$'s to $6$'s is obviously $1$.
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Assume a "fair" 6 sided die.
For N rolls, with a 6 on the Nth, by inspection
**0 <= 3's thrown <= (N-1)**
ie all throws before a 6 could gove one of [1 2 4 5] so no 3s,
or all throws before a 6 could produce 3's,
so (N-1) 3s are possible.
On any throw that does not produce a 6, the odds of throwing a 3 are 1 in 5 so you get "1/5th" of a 3 per throw, so the average number of 3's for N throws is [(N-1)/5] even though the actual number will be an integer on every occasion.
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There's a lot of answers here, but I thought I'd throw in an answer for both one and two dice. First let's consider only 3s and 6s. All other throws don't matter.
For one die: p(3|3 or 6) = 1/2 p(6|3 or 6) = 1/2
For two dice, there are exactly two ways to throw a 3 and exactly five ways to throw a 6: p(3|3 or 6) = 2/7 p(6|3 or 6) = 5/7
The probability of rolling n 3s before a 6 is p(3|3 or 6)^n.
Each additional 3 adds 1 to the expected count, leading to two infinite sums:
For one die: E[3 before 6] = 1*(1/2)^1 + 1*(1/2)^2 + 1*(1/2)^3 + ... = 1
For two dice: E[3 before 6] = 1*(2/7)^1 + 1*(2/7)^2 + 1*(2/7)^3 + ... = 2/5 = 0.4
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Define the random variables $X_i={\bf 1}(\text{roll $i$ is a $3$})$ and the stopping time $T=\{\text{index of first $6$}\}$. Observe that both $X_i$ and $T$ have finite expectation and that the $\{X_i\}$ are independent and identically distributed. Thus by Wald's identity, $$ {\bf E}(X_1+\cdots+X_T)={\bf E}X_1\cdot {\bf E}T. $$ Clearly ${\bf E}X_1=1/6$ and ${\bf E}T=6$. Thus the expected number of threes is $1$.
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Counting polynomials are fun.
A roll generates:
$$r = \frac{1}{6} x + \frac{1}{6} y + 4/6 r$$ or $$r = \frac{1}{2} x + \frac{1}{2} y$$
where x is the number of 3s and y is the number of 6s.
$$1 + r + r^2 + r^3 + ...$$ $$=\frac{1}{1-r}$$
We then take the derivative with respect to $x$ to get its count-weighted value, then evaluate at $x=1$:
$$\frac{d}{dx}\frac{1}{1-(\frac{x}{2}+\frac{y}{2})} = \frac{2 (y'(x)+1)}{(x+y-2)^2}$$
Now we want to evaluate at $y=0$ (as any case with a non-zero number of 6s should be discarded):
For $y`(x)$:
$$y = 2r-x$$ $$y'(x) = r'(x) - 1$$ $$r'(x) = 1/2$$ $$y'(x) = 1/2-1 = -1/2$$
So we get:
$$\frac{2 (-\frac{1}{2}+1)}{(1+0-2)^2}$$
$$=\frac{1}{1}$$
Or, the number of 3s is 1 on average.
This strategy can be generalized for more complex situations.
We can restrict ourselves to dice throws with outcomes $3$ and $6$. Among these throws, both outcomes are equally likely. This means that the index $Y$ of the first $6$ is geometrically distributed with parameter $\frac12$, hence $\mathbb{E}(Y)=2$. The number of $3$s occuring before the first $6$ equals $Y-1$ and has expected value $1$.