In the following experiment, determine whether or not the probabilities are binomial probabilities. If they are, identify the values for n, p, and q. A pair of dice are rolled 100 times. What is the probability of getting doubles x times?

Assuming the dice rolls are independent, the probability of getting doubles is [tex]\dbinom61\dfrac16\cdot\dfrac16=\dfrac6{36}=\dfrac16[/tex]. (That is, there are 6 ways of getting doubles, and the probability of getting a given face is 1/6.)

So the event of getting doubles in a set number of [tex]n[/tex] trials is binomially distributed with [tex]p=\dfrac16[/tex] (and so [tex]q=1-p=\dfrac56[/tex]).

You're given that [tex]n=100[/tex]. If [tex]X[/tex] is a random variable representing the number of doubles obtained in [tex]n[/tex] trials, then the probability of getting [tex]X=x[/tex] doubles is given by the PMF,

[tex]\mathbb P(X=x)=\begin{cases}\dbinom{100}x\dfrac16^x\cdot\left(\dfrac56\right)^{100-x}&\text{for }0\le x\le100\\\\0&\text{otherwise}\end{cases}[/tex]