Question

Suppose that an experiment can result in one of \(r\) possible outcomes, the ith outcome having probability \(p_{i}, i=1, \ldots, r, \sum_{i=1}^{r} p_{i}=1 .\) If \(n\) of these experiments are performed, and if the outcome of any one of the \(n\) does not affect the outcome of the other \(n-1\) experiments, then show that the probability that the first outcome appears \(x_{1}\) times, the second \(x_{2}\) times, and the \(r\) th \(x_{r}\) times is $$ \frac{n !}{x_{1} ! x_{2} ! \ldots x_{r} !} p_{1}^{x_{1}} p_{2}^{x_{2}} \cdots p_{r}^{x_{r}} \quad \text { when } x_{1}+x_{2}+\cdots+x_{r}=n $$ This is known as the multinomial distribution.

Short answer

The probability of the first outcome appearing \(x_1\) times, the second outcome appearing \(x_2\) times, and the rth outcome appearing \(x_r\) times equals: \[ \frac{n !}{x_{1} ! x_{2} ! \ldots x_{r} !} p_{1}^{x_{1}} p_{2}^{x_{2}} \cdots p_{r}^{x_{r}} \] when \(x_1 + x_2 + \cdots + x_r = n\), which is the multinomial distribution. We derived this by counting the number of ways to obtain the outcomes using the multinomial coefficient formula, calculating the probability of one specific sequence of outcomes, and then finding the total probability by multiplying these two results.

Step-by-step solution

Identifying the objective

Our goal is to show that the probability of the first outcome appearing \(x_1\) times, the second outcome \(x_2\) times, and so on up to the rth outcome appearing \(x_r\) times equals the formula: \[ \frac{n !}{x_{1} ! x_{2} ! \ldots x_{r} !} p_{1}^{x_{1}} p_{2}^{x_{2}} \cdots p_{r}^{x_{r}} \] provided that \(x_1 + x_2 + \cdots + x_r = n\).

Counting the number of ways to obtain the outcomes

In these experiments, we can obtain the outcomes in different orderings. To count the number of ways that we can obtain the outcomes, we can use the concept of combinations in combinatorics. Imagine that we have n slots, and in each slot, we write the outcome of the experiment. We can choose \(x_1\) slots for the first outcome, \(x_2\) slots for the second outcome, and so on up to \(x_r\) slots for the rth outcome. Thus, the number of ways to obtain these outcomes is equivalent to the number of different sequences that can be formed. This can be computed using the multinomial coefficient formula: \[ \frac{n !}{x_{1} ! x_{2} ! \ldots x_{r} !} \]

Calculating the probability of a specific sequence

Now we need to calculate the probability of one specific sequence of outcomes. Since each outcome is independent, we can multiply their probabilities. The probability of the first outcome appearing \(x_1\) times is \(p_{1}^{x_{1}}\), the second outcome appearing \(x_2\) times is \(p_{2}^{x_{2}}\), and so on up to the rth outcome appearing \(x_r\) times, which is \(p_{r}^{x_{r}}\). So, the probability of one specific sequence of outcomes is: \[ p_{1}^{x_{1}} p_{2}^{x_{2}} \cdots p_{r}^{x_{r}} \]

Calculating the total probability

As we have calculated the number of ways to obtain the outcomes and the probability of one specific sequence, we can find the total probability by multiplying these two results. The total probability is: \[ \frac{n !}{x_{1} ! x_{2} ! \ldots x_{r} !} p_{1}^{x_{1}} p_{2}^{x_{2}} \cdots p_{r}^{x_{r}} \] # Conclusion # We have shown that the probability of the first outcome appearing \(x_1\) times, the second outcome appearing \(x_2\) times, and the rth outcome appearing \(x_r\) times is equal to the formula mentioned in the exercise: \[ \frac{n !}{x_{1} ! x_{2} ! \ldots x_{r} !} p_{1}^{x_{1}} p_{2}^{x_{2}} \cdots p_{r}^{x_{r}} \] when \(x_1 + x_2 + \cdots + x_r = n\). This is the multinomial distribution.