Use the multinomial formula and find the probabilities for...

Use the multinomial formula and find the probabilities for each. $$ \begin{array}{l}{\text { a. } n=3, X_{1}=1, X_{2}=1, X_{3}=1, p_{1}=0.5, p_{2}=0.3} \\ {p_{3}=0.2} \\ {\text { b. } n=5, X_{1}=1, X_{2}=3, X_{3}=1, p_{1}=0.7, p_{2}=0.2} \\ {p_{3}=0.1} \\ {\text { c. } n=7, X_{1}=2, X_{2}=3, X_{3}=2, p_{1}=0.4, p_{2}=0.5} \\ {p_{3}=0.1}\end{array} $$

Use the multinomial formula and find the probabilities
for each.
$$
\begin{array}{l}{\text { a. } n=3, X_{1}=1, X_{2}=1, X_{3}=1, p_{1}=0.5, p_{2}=0.3} \\ {p_{3}=0.2} \\ {\text { b. } n=5, X_{1}=1, X_{2}=3, X_{3}=1, p_{1}=0.7, p_{2}=0.2} \\ {p_{3}=0.1} \\ {\text { c. } n=7, X_{1}=2, X_{2}=3, X_{3}=2, p_{1}=0.4, p_{2}=0.5} \\ {p_{3}=0.1}\end{array}
$$

Key Concepts

Multinomial Distribution

The multinomial distribution is a generalization of the binomial distribution that applies to experiments where each trial can result in more than two outcomes. It describes the probability of obtaining a specific combination of counts for each outcome over a fixed number of independent trials, each with its own constant probability of occurrence.

Multinomial Coefficient

The multinomial coefficient is a combinatorial factor given by n!/(x1!x2!...xr!), which counts the number of distinct ways to allocate n trials into r outcomes, where x1, x2, ..., xr represent the counts of each outcome. It plays a crucial role in calculating probabilities within the multinomial distribution.

Independence of Trials and Fixed Probabilities

A key assumption in the multinomial experiment is the independence of trials, meaning the outcome of one trial does not affect another. Additionally, each outcome has a fixed probability across all trials, which, when raised to the power corresponding to the count for that outcome and multiplied by the multinomial coefficient, provides the probability of observing a specific outcome combination.

Transcript

00:02 So this problem, we are going to use the multinomial formula, which reads p of x equals n factorial over x sub 1 factorial, times x sub 2 factorial, times x sub 3 factorial.

00:26 And depending on the number of events, you would keep that going until you get to your final event, which will call x sub k factorial.

00:35 And then we multiply that fraction by the probability of each of the events.

00:43 So we'd have the probability of event 1 raised to the x1 power, multiplied by the probability of event 2, raised to the x sub 2 power, multiplied by the probability of event 3, or raised to the power of x 3.

01:01 And again, we would keep that going until we got to the final event, and the probability of the final event will call p subk raised to the x subk power.

01:13 So in this particular problem, we know the value of n for part a to be three, so we would have three factorial in the top.

01:30 And x sub 1 is 1, so we'd have 1 factorial.

01:36 X sub 2 is also 1, so we'd have another 1 factorial, x sub 3 is also 1, so we'd have another 1 factorial in the bottom, and then we have to multiply by the probability of each of those events raised to their corresponding powers.

01:53 So p sub 1 is 0 .5, we use it to the x sub 1 power, p sub 2 is 0 .3, and we would raise it to the x sub 1 power, to the x sub 2 power, and p sub 3 is 0 .2, and we would raise it to the x sub 3 power.

02:19 And if you calculate that using your calculator, you will get a value of 0 .18 for the probability.

02:30 For part b, same formula, but in this instance, n is 5, so on the numerator, we'd have 5 factorial.

02:47 We have three events.

02:49 X sub 1 is 1.

02:53 X sub 2 is 3, and x sub 3 is 1...

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