Find the probability of a complementary event

A complementary event consists of all outcomes in the sample space that are not in a given event; the event and its complement are mutually exclusive and exhaustive, so their probabilities sum to 1, or 100%. For simple chance experiments, including equally likely outcomes, the probability of the complement is found as $1-P(A)$ and expressed as a fraction, decimal, or percentage; it is the probability of every outcome outside the event, not necessarily a single opposite outcome.

Detailed Explanation: Find the probability of a complementary event

A complementary event is the event that something does not happen. The event and its complement include every possible outcome, so their probabilities add to $1$ :

P(\text{complement of } A)=1-P(A)

Example

A fair number cube has sides numbered $1$ through $6$ . What is the probability of not rolling a number greater than $4$ ?

Let event $A$ be rolling a number greater than $4$ .

The possible outcomes are $5$ and $6$ , so:

P(A)=\frac{2}{6}=\frac{1}{3}

Use the complement rule:

P(\text{not }A)=1-\frac{1}{3}=\frac{2}{3}

Check the outcomes. The complement of rolling a number greater than $4$ is rolling $4$ or less: $1,2,3,$ or $4$ . That is $4$ outcomes out of $6$ :

\frac{4}{6}=\frac{2}{3}

Therefore, the probability of not rolling a number greater than $4$ is $\boxed{\frac{2}{3}}$ .

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Probability - Coins (2), Not All Specific, To Fraction

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