Question

Two experiments are to be performed. The first can result in any one of m possible outcomes. If the first experiment results in outcome i, then the second experiment can result in any of ni 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 outcome i, the second can end in $n_i$ways.

Step 2. Explanation.

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

If the first experiment ended in an outcome 1, the second can end in n₁ ways.

If the first experiment ended in an outcome 2, the second can end in n₂ ways.

If the first experiment ended in an outcome 3, the second can end in n₃ ways.

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

These are all possible outcomes of 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.