Probability of A or B: Subtract the Overlap

The probability of A or B is the chance that at least one of two events happens. In standard probability notation, “or” includes the possibility that both happen.

P(A or B) = P(A) + P(B) − P(A and B)

You may also see this written as P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The union symbol, ∪, means “or”; the intersection symbol, ∩, means “and.”

Why subtract? Adding the individual probabilities counts outcomes where both events happen twice. Subtracting their overlap leaves those outcomes counted once. This is the general addition rule.

One worked prediction-market example

Imagine a hypothetical contract asking whether a city will record rain or strong wind tomorrow. For this example, assume its rules define:

• A: At least 1 mm of rain at a named weather station.
• B: A maximum wind gust of at least 50 km/h at that same station.
• Yes: A happens, B happens, or both happen during the specified local calendar day, according to the named official weather report.

Your illustrative estimates—not live forecasts—are:

• P(A) = 40%
• P(B) = 30%
• P(A and B) = 20%

Then:

P(A or B) = 40% + 30% − 20% = 50%

The full breakdown lets you check the arithmetic:

Outcome	Probability
Rain only	40% − 20% = 20%
Strong wind only	30% − 20% = 10%
Both	20%
Neither	50%
Total	100%

The first three rows resolve Yes under our assumed rules. Together, they total 50%. Simply adding 40% and 30% would overstate the probability by 20 percentage points.

When can you simplify the formula?

Mutually exclusive events cannot both happen. Their overlap is zero, so:

P(A or B) = P(A) + P(B)

Independent events do not change each other’s probability. For those events:

P(A and B) = P(A) × P(B)

Therefore:

P(A or B) = P(A) + P(B) − P(A) × P(B)

These are different conditions, as the probability rules explain. Independence does not mean “no overlap.” In our example, assuming independence would give an overlap of 0.40 × 0.30 = 0.12 and an “or” probability of 58%, not 50%. That contradicts the stated 20% overlap estimate.

If you know only the 40% and 30% individual probabilities, you cannot determine one exact answer. The “or” probability could range from 40% to 70%: 40% if every B outcome is also an A outcome, or 70% if they never overlap. Don’t multiply merely because the overlap is unknown. A conditional probability estimate can help: P(A and B) = P(A) × P(B given A), when P(A) is positive.

Turn the probability into a contract check

First, verify what actually resolves Yes. Does “or” include both? Are the station, thresholds, time window and evidence source identical to those in your forecast? Polymarket, for example, says markets resolve according to their pre-defined rules, not simply a reader’s interpretation of the title.

Next, compare your estimate with an executable price, not just the displayed percentage. Polymarket’s price explanation dated March 13, 2026 says its display normally uses the bid–ask midpoint, switching to the last traded price when the spread exceeds 10 cents.

Continue the hypothetical example with these assumptions:

Item	Per Yes contract
Best bid	$0.45
Best ask	$0.49
Midpoint	$0.47
Illustrative total fee	$0.02
All-in purchase cost	$0.51

Assume one contract fills at the ask, pays $1 for Yes and $0 for No, and is held to settlement with no other costs. The fee is an arithmetic assumption, not a platform fee quote.

At your 50% estimate:

Expected payout = 0.50 × $1 = $0.50

Expected net result = $0.50 − $0.51 = −$0.01 per contract

The displayed 47% may look below your estimate, but the all-in cost implies a 51% break-even probability. Do not add the spread again: the purchase calculation already uses the ask.

Before risking money, check both the overlap assumption and the market’s rules and trading costs. Correct arithmetic does not make an uncertain forecast reliable.