probability - Fair dice throwing, expected value

We throw a standard fair dice until we threw $5$ and then $2,4$ or $6$ (not necessarily one after another). What is the expected value and variation of number of throws (let it be $X$)?

I was thinking about using the definition of expected value here, but to do that I need $\mathbb{P}(X=k)$ and I just have no clue how to easily express that probability.

Answer

Let $X$ be the random number of rolls needed to obtain the first $5$, and let $Y$ be the random number of rolls needed to obtain the first instance of $2$, $4$, or $6$ after the first $5$. Then $Z = X+Y$ is the random variable for the total number of rolls needed to observe the event of interest. Clearly, $X$ and $Y$ are independent, so $\mathrm{E}[Z] = \mathrm{E}[X] + \mathrm{E}[Y]$ and $\mathrm{Var}[Z] = \mathrm{Var}[X] + \mathrm{Var}[Y]$. We then need only characterize the distributions of $X$ and $Y$. To this end, note that $X$ must be a geometric random variable: for each die roll is independent of any previous die roll, and the probability of obtaining a $5$ on a given die roll is constant, $p = 1/6$. Thus

$$\Pr[X = x] = (1-p)^{x-1}p, \quad x = 1, 2, 3, \ldots.$$ Similarly, $Y$ is also geometric, but with probability $\pi = 1/2$, since the probability of rolling any even number on a single roll is $3/6 = 1/2$. From this information, it is straightforward to obtain the mean and variance of $Z$.