How can I calculate the chance that either of two events happens without double-counting?

Check whether the events share any possible outcomes. If they do, those shared outcomes are included once in P(A) and again in P(B), so adding the two probabilities counts them twice. The addition rule is P(A or B) = P(A) + P(B) − P(A and B): subtract the probability of the overlap once. “A or B” here means A happens, B happens, or both happen. If the events cannot occur together, P(A and B) is zero and you can add directly. [S1]

A simple diagram is:

A only + shared outcomes + B only ↑ counted in both A and B

Adding P(A) and P(B) counts the shared section twice; subtracting P(A and B) leaves it counted once.

Suppose one fair six-sided die is rolled, with the six outcomes 1, 2, 3, 4, 5, and 6 treated as equally likely. Let A mean “the result is at most 3,” and B mean “the result is odd.”

A = {1, 2, 3} B = {1, 3, 5} Shared outcomes, A and B = {1, 3} Either event, A or B = {1, 2, 3, 5}

There are four outcomes in the union out of six, so P(A or B) = 4/6 = 2/3. Using the rule gives the same result: P(A) = 3/6, P(B) = 3/6, and P(A and B) = 2/6. Thus 3/6 + 3/6 − 2/6 = 4/6.

The lists make the overlap visible: 1 and 3 appear in both events, so they must be removed once from the initial sum. This calculation assumes the six outcomes are equally likely; the counting method would need adjustment if that assumption did not hold.

Use the same rule, but you need the overlap probability as well as the two individual probabilities. For example, with illustrative figures only, suppose P(A) is 40%, P(B) is 30%, and P(A and B) is 10%. Then P(A or B) = 40% + 30% − 10% = 60%.

If the overlap isn’t supplied, look for enough information to calculate it, such as a count of cases that satisfy both conditions. If you have neither the overlap nor information that establishes it is zero, the two individual percentages alone generally do not give one exact answer. Don’t silently assume the events are separate: first check whether one outcome or case could satisfy both.

Use this fill-in template:

1. Event A means: [define A]. Event B means: [define B].
2. P(A) = [probability of A].
3. P(B) = [probability of B].
4. P(A and B) = [probability both happen].
5. P(A or B) = [A] + [B] − [both] = [result].

For a small outcome set, write A and B as sets, find the outcomes appearing in both, and count the distinct outcomes in either set. Then compare that count with the formula. If the events are mutually exclusive, the shared set is empty and the overlap term is zero. [S1]

As a final check, the answer should not count any shared case twice. The rule tells you how to combine probabilities when you know the overlap; it does not establish whether particular real-world events overlap or supply missing probabilities. Those details must come from the problem’s definitions or data.