Distribution of the sum of the number of rolls that result in an even number

Question

I'd like to ask for some help in finding the distribution of the number of rolls that result in an even number. The problem goes like this:

Consider a loaded die where the probability that a roll results in $I$ is $p_i = I/21$, $i = 1,\ldots,6$. Suppose the die is rolled $35$ times and let $N_i$ be the number of times that the roll results in $i$. Then $N = (N_1, \ldots , N_6)$ is multinomial distributed with parameters $35$, $\left(\frac1{21}, \ldots, \frac6{21}\right)$, find the distribution of $X = N_2+N_4+N_6$.

In my attempt, knowing that $N_i$'s follow a $\text{Binomial}(35,p_i)$ distribution, I conditioned on $N_6$, and then on $ N_4$ but my expression was too complicated to simplify. I was wondering if there is an easier way to approach this problem? This is just a practice problem as I study for my exam.

Answer 1

Hint:

• What is the probability a particular single roll is even?
• Are rolls independent?
• What is the probability exactly $X$ rolls are even out of $35$ total rolls?