Calculate the probability of two or three events both occurring — independent or dependent — with breakdowns, odds, and repeated trial analysis.
The AND probability calculator computes the probability that two (or three) events both occur — their intersection. This is one of the most fundamental operations in probability theory and is governed by the multiplication rule: for independent events P(A ∩ B) = P(A) × P(B), and for dependent events P(A ∩ B) = P(A) × P(B|A).
Understanding when to apply each version is critical. Drawing two aces from a standard deck without replacement involves dependent events because the first draw changes the deck composition. Flipping a coin and rolling a die are independent because one outcome doesn't influence the other.
This calculator supports both modes, lets you extend to three events, and provides a complete breakdown showing the probability of each possible outcome combination. A repeated-trials table shows how the probability accumulates over multiple independent opportunities, which is useful for reliability engineering, quality control, and game strategy. Check the example with realistic values before reporting.
Calculating AND probability is essential in risk assessment, reliability engineering, medical testing, insurance, and everyday decision-making. Whenever you need to know the chance that multiple conditions are all met simultaneously, you need the multiplication rule.
This tool removes the guesswork and handles both independent and dependent cases — a distinction many people get wrong, leading to significantly incorrect risk estimates.
Independent: P(A ∩ B) = P(A) × P(B). Dependent: P(A ∩ B) = P(A) × P(B|A). Three events (independent): P(A ∩ B ∩ C) = P(A) × P(B) × P(C).
Result: 0.15 (15%)
For two independent events with P(A) = 0.5 and P(B) = 0.3, the probability both occur is 0.5 × 0.3 = 0.15 or 15%.
The multiplication rule states that the probability of two events both occurring equals the product of their individual probabilities — but only when they're independent. When events are dependent, you must use the conditional version P(A ∩ B) = P(A) × P(B|A). Confusing these two cases is one of the most common errors in applied probability.
In engineering, system reliability is calculated by multiplying component reliabilities (for series systems). A system with two components, each 99% reliable, has overall reliability of 0.99 × 0.99 = 98.01%. In medical testing, the probability of two independent symptoms co-occurring helps with differential diagnosis. In finance, the joint probability of multiple market conditions drives portfolio risk models.
People often assume independence when events are actually dependent. For example, drawing cards without replacement creates dependency. Another mistake is confusing "and" with "or" — they answer fundamentally different questions and use different formulas.
AND (intersection) is the probability both events occur. OR (union) is the probability at least one occurs. AND always gives a smaller or equal result than OR.
Events are independent when the occurrence of one doesn't affect the probability of the other. Coin flips, separate dice rolls, and independent sensor failures are common examples.
No. P(A ∩ B) ≤ min(P(A), P(B)). Both events happening can't be more likely than the less likely event alone.
For independent events, simply multiply all individual probabilities. For dependent events, you need the full chain: P(A) × P(B|A) × P(C|A∩B) × ...
Mutually exclusive events can never both occur, so P(A ∩ B) = 0. If you get zero and expected otherwise, check whether the events truly can co-occur.
Enter values between 0 and 1. To convert a percentage, divide by 100: for example, 35% becomes 0.35.