Expand Your Knowledge: Multinomial Probability Distribution Consider
a multinomial experiment. This means the following:
1. The trials are independent and repeated under identical conditions.
2. The outcomes of each trial falls into exactly one of $k \geq 2$ categories.
3. The probability that the outcomes of a single trial will fall into ith category is $p_{i}$ (where $i=1,2 \ldots, k$ ) and remains the same for each trial. Furthermore, $p_{1}+p_{2}+\ldots+p_{k}=1$
4. Let $r_{i}$ be a random variable that represents the number of trials in which the outcomes falls into category $i$. If you have $n$ trials, then $r_{1}+r_{2}+\ldots$ $+r_{k}=n .$ The multinational probability distribution is then $$P\left(r_{1}, r_{2}, \cdots r_{k}\right)=\frac{n !}{r_{1} ! r_{2} ! \cdots r_{2} !} p_{1}^{r_{1}} p_{2}^{(2)} \cdots p_{k}^{r_{2}}$$
How are the multinomial distribution and the binomial distribution related? For the special case $k=2,$ we use the notation $r_{1}=r, r_{2}=n-r, p_{1}=p$ and $p_{2}=q .$ In this special case, the multinomial distribution becomes the binomial distribution.
The city of Boulder, Colorado is having an election to determine the establishment of a new municipal electrical power plant. The new plant would emphasize renewable energy (e.g., wind, solar, geothermal). A recent large survey of Boulder voters showed $50 \%$ favor the new plant, $30 \%$ oppose it, and $20 \%$ are undecided. Let $p_{1}=0.5, p_{2}=0.3,$ and $p_{3}=0.2 .$ Suppose a random sample of $n=6$ Boulder voters is taken. What is the probability that
(a) $r_{1}=3$ favor, $r_{2}=2$ oppose, and $r_{3}=1$ are undecided regarding the new power plant?
(b) $r_{1}=4$ favor, $r_{2}=2$ oppose, and $r_{3}=0$ are undecided regarding the new power plant?