"Multinomial Experiments In Exercises 39 and 40, use the infor...

Question

"Multinomial Experiments In Exercises 39 and 40, use the information below.

A multinomial experiment satisfies these conditions.

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:

Each outcome has a fixed probability. So, . The sum of the probabilities for all outcomes is

The number of times occurs is , the number of times occurs is , the number of times occurs is , and so on.

The discrete random variable x counts the number of times that each outcome occurs in n independent trials where . The probability that x will occur is

Genetics Another proposed theory in genetics gives the corresponding probabilities for the four types of plants described in Exercise 39 as , and . Ten plants are selected. Find the probability that 5 will be tall and colorful, 2 will be tall and colorless, 2 will be short and colorful, and 1 will be short and colorless."

Accepted Answer

All right, hello, everyone. So this question says, a fast food restaurant offers 4 types of meals, burger meal, chicken meal, vegetarian meal, and fish meal. Based on past data, the probability that a customer orders each type is as follows. 12 customers placed their orders independently. What is the probability that exactly 4 order burger meals, 3 order chicken meals, 4 order vegetarian meals, and 1 orders a fish meal? And here we have 4 different answer choices labeled A through D. All right, so first, let's write out the information that we know. In this case, N, which is the number of trials, is equal to 12. And we know not only the meal types that were given, but also their probabilities and desired outcomes. The probability of a burger meal or P1 is 3/10. And the desired outcome for burger orders or X of 1 is 4. P2 for chicken meal is 2/10. And X2 is equal to 4, excuse me, equal to 3. P3 for the vegetarian meal is 4/10. And X3. Is 4. Lastly, P4 for the fish meal is 1 out of 10. And X4 is equal to 1. So for this question, we're going to use the multinomial formula. So P of X, which is the probability of this specific outcome. Is equal to. And factorial Divided by X1 factorial, multiplied by X2 factorial, multiplied by X3 factorial, multiplied by X4 factorial. And this is multiplied then. By piece of one to the power of X1. Multiplied by P2 to the power of X2. Multiplied by P3 to the power of X3. And multiplied by P4 to the power of X4. So now, let's substitute the values given. So P Is equal to 12 factorial. Divided by 4 factorial, multiplied by 3 factorial, multiplied by 4 factorial, multiplied by 1 factorial. This is then further multiplied by. 3/10 to the 4th power. Multiplied by. 2/10 multiplied to the 3 power. Multiplied by 4 out of 10, the 4th power. And multiplied by 1/10 to the first power. Ultimately Evaluating this expression gives you a value of P equal to 0.0230. And this Corresponds to option D in the multiple choice. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

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