Probability Calculator · Practical guide
Probability of Independent Events
When one event does not change the other's odds, multiply the probabilities to get P(A and B).
At a glance
The Multiplication Rule
Two events are independent when one outcome tells you nothing about the other — separate coin flips, dice rolls, or draws with replacement. For independent events, the chance both happen is the product: P(A AND B) = P(A) × P(B). Two heads in a row: 0.5 × 0.5 = 0.25, a 25% chance.
Interactive tool
Probability Calculator
Enter your own values, then compare the result with the examples below.
A Worked Example
Suppose event A has probability 0.6 and independent event B has probability 0.5:
| Quantity | Formula | Result |
|---|---|---|
| P(A AND B) | 0.6 × 0.5 | 0.30 |
| P(A OR B) | 0.6 + 0.5 − 0.30 | 0.80 |
Why OR Needs the Subtraction
Adding 0.6 + 0.5 double-counts the outcomes where both events happen. Subtracting the overlap once — the addition rule, P(A OR B) = P(A) + P(B) − P(A AND B) — corrects that, keeping the answer a valid probability between 0 and 1. The calculator applies both formulas automatically when you select the Independent relationship.
How to Tell Events Are Independent
Ask whether one outcome changes the other's odds. Coin flips and dice: yes, independent. Cards drawn without replacement: no — removing an ace changes the deck. When independence fails, switch to the conditional formula P(A AND B) = P(A) × P(B | A) instead.
Frequently Asked Questions
What is the probability of flipping two heads in a row?
0.5 × 0.5 = 0.25, or 25%. Each flip is independent, so the probabilities multiply.
Does order matter for P(A and B)?
No. Multiplication is commutative, so P(A) × P(B) equals P(B) × P(A) — the chance of both events occurring is the same either way.
What if the events are not independent?
Use the conditional rule P(A AND B) = P(A) × P(B | A), where P(B | A) is the probability of B after A has already happened.
Can P(A or B) ever exceed 1?
No. The addition rule subtracts the overlap P(A AND B), which guarantees the result stays within 0 to 1 — the calculator handles this for you.