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Worked example

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Addition Rule of Probability

A public probability question that shows why the probability of an overlap must be subtracted when combining two non-mutually-exclusive events.

Question

In a commuter survey, the probability that a randomly chosen respondent rides the bus is 0.450.45, the probability that the respondent cycles is 0.300.30, and the probability that the respondent does both is 0.150.15. What is the probability that the respondent rides the bus or cycles?

  1. a.0.1350.135
  2. b.0.150.15
  3. c.0.600.60
  4. d.0.750.75

Correct answer

C. 0.600.60

Full reasoning

  1. 1The events overlap because some respondents both ride the bus and cycle, so they are not mutually exclusive.
  2. 2Apply the addition rule: P(bus or cycle)=P(bus)+P(cycle)−P(both)P(\text{bus or cycle}) = P(\text{bus}) + P(\text{cycle}) - P(\text{both}).
  3. 3Substitute the values: 0.45+0.30−0.15=0.600.45 + 0.30 - 0.15 = 0.60.

Why each choice is right or wrong

Choice A

This multiplies 0.45×0.300.45 \times 0.30, which would estimate the probability of doing both under independence, not the probability of either event.

Choice B

This reports the given overlap P(both)P(\text{both}) instead of combining the three probabilities with the addition rule.

Choice C

Correct. The addition rule gives 0.45+0.30−0.15=0.600.45 + 0.30 - 0.15 = 0.60, removing the double-counted respondents who do both.

Choice D

This adds 0.45+0.300.45 + 0.30 as if the events were mutually exclusive, double-counting the respondents who both ride the bus and cycle.

Related formula

Addition rule of probability

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

Assumptions: All probabilities describe the same random selection from one population.

Related topic and practice

Descriptive StatisticsProbability and CountingCombinationsMedian of a Data SetTwo draws without replacementZ-Scores on a Normal DistributionCreate a free account to practice

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Sources

Authored for this public explainer registry. It has no database question ID and is not copied from a protected or official exam bank.

  1. Introductory Statistics 2e, Section 3.3: Two Basic Rules of Probability — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
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Sources last checked 2026-08-07

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