Determine probabilities of complementary events

An event and its complement consist of all outcomes in the sample space divided into two mutually exclusive, exhaustive sets: the event occurs or it does not. Given the probability of one, the probability of the complement is found using $P(A^c)=1-P(A)$ , with 1 representing certainty, and this relationship is applied across fractions, decimals, percentages, tables, and simple probability models; the complement is not the reciprocal of the original probability.

Detailed Explanation: Determine probabilities of complementary events

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(A)+P(A^c)=1

Therefore, use:

P(A^c)=1-P(A)

Example

The probability that it will rain tomorrow is $\frac{3}{10}$ . What is the probability that it will not rain?

Let $A$ be the event “it rains.” Then:

P(A)=\frac{3}{10}

The complement, $A^c$ , is “it does not rain.”
Subtract the probability of rain from $1$ :

P(A^c)=1-\frac{3}{10}

Write $1$ as $\frac{10}{10}$ and subtract:

P(A^c)=\frac{10}{10}-\frac{3}{10}=\frac{7}{10}

So, the probability that it will not rain is:

\boxed{\frac{7}{10}}

Remember, the complement is found by subtracting from $1$ —it is not the reciprocal of the original probability.

Learn by doing: Determine probabilities of complementary events

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

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

Earned ?