Determine probabilities of mutually exclusive events

Mutually exclusive events are outcomes or sets of outcomes that cannot occur in the same trial, so the probability that one or the other occurs is the sum of their probabilities: $P(A\text{ or }B)=P(A)+P(B)$ . The understanding applies to finite sample spaces represented with lists, tables, or diagrams and includes recognizing that overlapping events must not be added directly; conditional, continuous, and more generalized probability treatments are beyond this scope.

Detailed Explanation: Determine probabilities of mutually exclusive events

Mutually exclusive events are events that cannot happen at the same time in one trial.

For mutually exclusive events:

P(A\text{ or }B)=P(A)+P(B)

Example

A fair six-sided die is rolled. What is the probability of rolling a $2$ or a $5$ ?

Step 1: Check whether the events are mutually exclusive.

A die can show only one number on a single roll. It cannot show both $2$ and $5$ at the same time, so these events are mutually exclusive.

Step 2: Find each probability.

There is $1$ way to roll a $2$ out of $6$ possible results:

P(2)=\frac{1}{6}

Similarly,

P(5)=\frac{1}{6}

Step 3: Add the probabilities.

P(2\text{ or }5)=\frac{1}{6}+\frac{1}{6} =\frac{2}{6} =\frac{1}{3}

So, the probability of rolling a $2$ or a $5$ is:

\boxed{\frac{1}{3}}

Only add probabilities directly when the events cannot overlap.

Learn by doing: Determine probabilities of mutually exclusive events

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