Outcome

By Jude Wallis · Updated

An outcome is a single possible result of one trial, and the collection of every outcome for that trial is the sample space.

An outcome is one complete result of a single trial and cannot be broken into smaller results. An event is any set of outcomes, so events are assembled out of outcomes and never the reverse, and an event's probability is the total probability of the outcomes inside it. What counts as an outcome is fixed by what you record on each trial, and that choice does most of the damage below.

Roll two fair dice and record both faces. The sample space holds 6×6=366 \times 6 = 36 ordered outcomes, each carrying probability 1/360.02781/36 \approx 0.0278 by symmetry. Six of them total 7, running from (1,6)(1,6) to (6,1)(6,1), so P(sum of 7)=6/360.1667P(\text{sum of } 7) = 6/36 \approx 0.1667. Exactly one totals 2, so P(sum of 2)=1/360.0278P(\text{sum of } 2) = 1/36 \approx 0.0278.

Now the misreading: there are 11 possible sums, 2 through 12, so each has probability 1/110.09091/11 \approx 0.0909. Those 11 sums are a legitimate way to describe a roll, but they are events rather than outcomes, and they are events of very different sizes. A sum of 7 collects six outcomes while a sum of 2 collects one, so the first is six times as likely as the second, not equal to it. Counting only produces a probability across items that carry equal probability, and grouping outcomes into categories is the quickest way to lose that.

Outcomes are not required to be equally likely, and nothing in the definition promises they will be. A loaded die still has six outcomes; you add their stated probabilities rather than counting them. What does always hold across a finite sample space is that the outcome probabilities total 1.

Topic 2.4, Introduction to Probability, defines the sample space as the set of all possible nonoverlapping outcomes. Nonoverlapping is the load-bearing word. If two items on your list could both be true of the same trial, you have written down events, and the counting shortcut no longer applies to them.

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