Probability Rules: Check “Not,” “Or” and “And” Before the Price

Probability rules tell you how to combine forecasts without double-counting outcomes or wrongly assuming events are independent. In prediction markets, they help distinguish three very different questions: will an event happen, will either event happen, and will both happen?

They check whether your estimates fit together. They do not establish whether those estimates are accurate—or whether a contract is worth its cost.

Start with the event, not the number

Before calculating, define exactly what counts as Yes. Polymarket’s resolution guide, for example, says markets settle according to their predefined rules, with winning shares receiving $1 and losing shares becoming worthless. The headline alone is not the settlement test.

For one hypothetical example, define:

• A: District A’s official certified election result names Party X’s candidate as winner by the specified deadline.
• B: District B’s official certified election result names Party X’s candidate as winner by the specified deadline.

For this exercise, every other case counts as No, with no cancellation or alternative payout. These are invented terms, not an actual market’s rules. Below, “wins” means satisfying that certification-and-deadline test.

Suppose your forecasts produce this table:

Outcome	Assumed probability
Party X wins both districts	40%
Wins A only	20%
Wins B only	10%
Wins neither	30%
Total	100%

The four rows are mutually exclusive: only one can happen. They are also exhaustive: together they cover every possibility under our assumptions.

From the table, P(A) = 60% and P(B) = 50%. Here, P means “probability of.” All the numbers are teaching assumptions, not election forecasts.

The core probability rules

1. Probabilities stay between 0 and 1

A probability must be between 0 and 1, equivalent to 0%–100%. Mutually exclusive outcomes covering all possibilities must sum to 1. These follow from the basic probability axioms described in Penn State’s probability lesson.

Our table passes that check: 40% + 20% + 10% + 30% = 100%.

Two possible outcomes do not imply a 50/50 chance. Yes and No can be the only outcomes while having very different probabilities.

2. “Not”: use the complement rule

P(not A) = 1 − P(A)

If Party X has a 60% chance of winning A, its chance of not winning A is:

1 − 0.60 = 0.40, or 40%.

The complement rule applies to an event and everything outside it. For a contract, verify that No really covers every case outside Yes; do not ignore special settlement provisions.

3. “Or”: add, then subtract the overlap

P(A or B) = P(A) + P(B) − P(A and B)

In probability, “or” includes the case where both happen. Party X’s chance of winning at least one district is:

0.60 + 0.50 − 0.40 = 0.70, or 70%.

Simply adding 60% and 50% would give 110% because the both-win outcome was counted twice. Subtracting the overlap fixes that. For mutually exclusive events, the overlap is zero, so the addition rule simplifies to adding their probabilities.

“Exactly one” is different: our table gives 20% + 10% = 30%.

4. “And”: multiply using a conditional probability

P(A and B) = P(A) × P(B given A), provided P(A) > 0.

The table assigns 60% probability to A, including 40% to both wins. Restricting attention to the cases where A happens gives:

P(B given A) = 0.40 ÷ 0.60 ≈ 66.7%.

Multiplying 0.60 by 0.40/0.60 gives 0.40, our both-win probability. This is the general multiplication rule.

The shortcut P(A and B) = P(A) × P(B) works only for independent events—when knowing one occurred does not change the probability of the other. Here, B’s conditional probability is about 66.7%, not its unconditional 50%, so the events are not independent. The shortcut would incorrectly give 30% instead of 40%.

Mutually exclusive and independent are not synonyms: the first means the events cannot both happen; the second means one does not change the probability of the other. For more on the meaning of “given,” see probability and conditional probability.

5. A narrower event cannot be more likely

“Both districts” is contained within “District A.” Its probability cannot exceed A’s probability—or B’s. This follows from the subset rule.

Our 40% both-win estimate is below both 60% and 50%. An estimate of 55% for both would contradict the 50% estimate for B.

Then compare the forecast with executable cost

Suppose a hypothetical $1-or-$0 contract pays Yes if Party X wins at least one district under the terms above. Your calculated probability is 70%.

Assume the Yes bid is 67¢, the ask is 71¢, and ten shares are available at that ask. Also assume you buy ten shares, hold through settlement, pay a fixed total purchase fee of 10¢, and incur no further fees. These are illustrative quotes and costs, not a platform fee schedule.

• Spread: 71¢ − 67¢ = 4¢.
• Purchase cost: 10 × $0.71 + $0.10 = $7.20.
• Expected payout: 70% × $10 + 30% × $0 = $7.00.
• Expected net result: $7.00 − $7.20 = −$0.20.

The break-even probability is $7.20 ÷ $10 = 72%, above your 70% estimate. The spread is already reflected in buying at the ask; do not add it again as a separate fee. The expected result is not the settlement result: this purchase would gain $2.80 on Yes or lose $7.20 on No.

A displayed percentage may not be your entry price. As of October 11, 2026, Polymarket’s price guide says its display uses the bid-ask midpoint, or the last traded price when the spread exceeds 10¢.

Before any order, match the forecast to the resolution wording, check overlap and dependence, then compare it with the full cost at your order size. Use the broader pre-trade checklist for liquidity and settlement checks. Correct arithmetic is a consistency check, not a reason by itself to risk money.