EZ Math

Statistics · Probability

How to Find the Probability of the Complement

Problem

P(rain)=0.3P(\text{rain}) = 0.3

Answer

P(no rain)=0.7P(\text{no rain}) = 0.7

The complement is everything the event does not cover. Its probability is always 1 minus the probability of the event.

Step-by-step solution

  1. Write the complement rule.

    P(not A)=1−P(A)P(\text{not } A) = 1 - P(A)

    Why: An event either happens or it does not, so the two probabilities fill the whole sample space.

  2. Substitute the known probability.

    P(no rain)=1−0.3P(\text{no rain}) = 1 - 0.3

    Why: No rain is the complement of rain.

  3. Subtract.

    1−0.3=0.71 - 0.3 = 0.7

    Why: The two probabilities add to 1.

Common mistakes

  • Subtracting from 0 instead of 1.
  • Subtracting the complement from the event, which flips the answer.
  • Assuming the two outcomes are equally likely. The complement rule works for any probability.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

  1. P(A)=15P(A) = \frac{1}{5}

    Show answer

    P(not A)=45P(\text{not } A) = \frac{4}{5}

  2. P(A)=0.62P(A) = 0.62

    Show answer

    P(not A)=0.38P(\text{not } A) = 0.38

  3. P(A)=38P(A) = \frac{3}{8}

    Show answer

    P(not A)=58P(\text{not } A) = \frac{5}{8}

  4. P(A)=25%P(A) = 25\%

    Show answer

    P(not A)=75%P(\text{not } A) = 75\%

  5. P(A)=0.09P(A) = 0.09

    Show answer

    P(not A)=0.91P(\text{not } A) = 0.91