Question

Two experiments are to be performed. The first can result in any one of m possible outcomes. If the first experiment results in an outcome$i,$then the second experiment can result in any of$n_i$ the possible outcomes, i = 1, 2, ..., m. What is the number of possible outcomes of the two experiments?

Short answer

$n_1+n_2+n_3+\dots +n_m$

Use the principle of the sum. Divide the result of experiment 1.

Step-by-step solution

Given Information.

2 Experiments.

The first can end in $m$outcomes.

If the first experiment ends in an outcome $i,$the second can end in $n_i$ways.

Explanation.

Just count the experiment's results and divided them into sections according to the first experiment result. That's called sometimes the principle sum.

If the first experiment ended in an outcome $1,$the second can end in $n_1$ways

If the first experiment ended in an outcome $2,$the second can end in $n_2$ways

If the first experiment ended in an outcome $3,$the second can end in $n_3$ways

If the first experiment ended in an outcome $m$, the second can end in$n_m$ways

These are all possible outcomes of the experiment $1$, and only one of them can happen. The total number of outcomes is$n_1+n_2+n_3+\dots +n_m$.

Conclusion.

$n_1+n_2+n_3+\dots +n_m$

By using the sum principle to divide according to the result of the given experiment 1.