Problem 5 (35 pts) Multinomial Experiments A multinomial Experiment meets the following criteria: . The experiment has a fixed number of trials n, where each trial is independent of the other trials. . Each trial has k possible mutually exclusive outcomes En, Ez Ex..., Ex. . Each outcome has a fixed probability. So, P(E,)=PL, P(Ez)=P2 P(E3)=py...,P(Ex)=px. The sum of the probabilities for all outcomes is pitpz+p3+...+pk=1. The number of times E, is X1. The number of times Ez is X2. The number of times Es is x3. And so on. . The discrete random variable x counts the number of times XL,X2X3...xx that each outcome occurs in n independent trials where XitX2+X3+...+X=n. The probability that x will occur is n! P(x) = "X1!xz! X3! ... xx!P1 P2 P3 .Pk 1. According to a theory in genetics, when tall and colorful plants are crossed with short and colorless plants, four types of plants will result: tall and colorful, tall and colorless, short and colorful, and short and colorless, with corresponding probabilities of 9/16, 3/16, 3/16 and 1/16. Twelve plants are selected. Find the probability that 6 will be tall and colorful, 3 will be tall and colorless, 2 will be short and colorful and 1 will be short and colorless. 2. Another proposed theory in genetics gives the corresponding probabilities for the four types of plants described in part 1 as 5/16, 4/16, 3/16 and 6/16. Twelve plants are selected. Find the probability that 6 will be tall and colorful, 3 will be tall and colorless, 2 will be short and colorful and 1 will be short and colorless