Addition Rule of Probability: Public Worked Question

Question

In a commuter survey, the probability that a randomly chosen respondent rides the bus is $0.45$ , the probability that the respondent cycles is $0.30$ , and the probability that the respondent does both is $0.15$ . What is the probability that the respondent rides the bus or cycles?

a. $0.135$
b. $0.15$
c. $0.60$
d. $0.75$

Correct answer

C. $0.60$

Full reasoning

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

Why each choice is right or wrong

Choice A

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

Choice B

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

Choice C

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

Choice D

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

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