Alissa Hancock

2022-07-07

Suppose we perform 2 experiments and this first experiment can result in any one of $m$ possible outcomes. Suppose the first experiment results in outcome $i$. Then, the second experiment can result in any of ${n}_{i}$ possible outcomes $i=1,..,m$. $\mathbf{Q}\mathbf{s}\mathbf{:}$ what is the number of possible outcomes of the two experiments?

Attempt:
If we have $1,...,m$ possible outcomes and say this experiment results in one of them say $i$. and so second experiment can result in ${n}_{i}$ for each $i$ so by multiplication principle we have ${n}_{1}\cdot {n}_{2}\cdot ..\cdot {n}_{m}$ possible outcomes. Is this correct?

Jasper Parsons

Expert

To have the outcomes implied by ${n}_{1}$ in the second experiment, we need outcome 1 in the first experiment (${E}_{1}=1$). Similarly we need outcome ${E}_{1}=2$ to get to those ${n}_{2}$ possibilities, etc.
So the possible second-experiment outcomes each depend on a given first-experiment result and the total outcome options are $\sum _{1}^{m}\left(1\cdot {n}_{i}\right)=\sum _{1}^{m}{n}_{i}$

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