Question

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?

Answer 1

Assuming the dice rolls are independent, the probability of getting doubles is $\dbinom61\dfrac16\cdot\dfrac16=\dfrac6{36}=\dfrac16$. (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 $n$ trials is binomially distributed with $p=\dfrac16$ (and so $q=1-p=\dfrac56$).

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

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