The different probabilites, binomial distribution?

Question

Assume that the probability of a citizen walking, using a bicycle or taking the bus to work is $0.65, 0.25, 0.1$ respectively.

If we choose 7 random citizens, what is the probability of $1$ walking, $3$ using a bicycle and $3$ taking the bus?

I'm currently learning about binomial, hypergeometric, poisson and negative binomial distributions. I'm not sure which one this one fits under, as I have 3 different probabilities for three different situations. Any ideas?

Answer 1

There is a distribution known as the multinomial distribution, it can be viewed as a generalization of the binomial distribution.

We have

\begin{align}Pr(X_1 = x_1, X_2 = x_2, X_3=x_3) &= \binom{x_1+x_2+x_3}{x_1, x_2, x_3}\prod_{i=1}^3p_i^{x_i}\\&=\frac{(x_1+x_2+x_3)!}{x_1!x_2!x_3!}\prod_{i=1}^3p_i^{x_i}\end{align}