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 ≥ 2 categories.

3. The probability that the outcomes of a single trial will fall into ith category is pi (where i = 1, 2…, k) and remains the same for each trial.

Furthermore, p₁ + p₂ + … + 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 r₁ + r₂ + … + r_k = n. The multinomial probability distribution is then

How are the multinomial distribution and the binomial distribution related? For the special case k = 2, we use the notation r₁ = r, r₂ = n – r, p₁ = p, and p₂ – 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 p₁ – 0.5, p₂ – 0.3, and p₃ – 0.2. Suppose a random sample of n – 6 Boulder voters is taken. What is the probability that

(a) r₁ – 3 favor, r₂ – 2 oppose, and r₃ – 1 are undecided regarding the new power plant?

(b) r₁ – 4 favor, r₂ – 2 oppose, and r₃ – 0 are undecided regarding the new power plant?

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