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Probability Calculator: Combine Independent Events

This free probability calculator combines the chances of two independent events. It finds the probability of both, either, neither or exactly one, and the chance of at least one success over repeated tries.

By Debmedia Technologies LLP · Kolkata, India · Updated

What Is It and When Do You Need It?

Probability measures how likely something is, from 0 (impossible) to 1 (certain). Questions about combining chances come up in games, risk, quality control and everyday decisions.

Enter the chances of two independent events and, if you like, a number of repeated tries.

How to Use This Tool

  1. Enter P(A) as a decimal or a percentage.
  2. Enter P(B).
  3. Optionally enter the number of repeated trials.
  4. Press Calculate.

How the Calculation Works

For independent events: P(A and B) = P(A) × P(B), P(A or B) = P(A) + P(B) − P(A and B), P(neither) = (1 − P(A)) × (1 − P(B)) and P(exactly one) = P(A)(1 − P(B)) + P(B)(1 − P(A)). For n repeated tries of A, P(at least one) = 1 − (1 − P(A))^n.

Worked Example

With P(A) = 0.5 and P(B) = 0.3, both happen with probability 0.15, either with 0.65, neither with 0.35 and exactly one with 0.5.

Common Uses

  • Games and cards.
  • Risk and reliability.
  • Quality control.
  • Homework.

Tips and Common Mistakes

  • Check that events are really independent.
  • Percent signs are accepted.
  • Use the binomial calculator for exact counts of successes.

Next, you may need the binomial calculator, permutation and combination calculator, normal distribution calculator, sample size calculator.

Everything runs in your browser. Nothing you type is uploaded, stored or sent to any outside service.

Frequently Asked Questions

Two events are independent if one happening does not change the chance of the other, such as two separate coin flips.
Multiply their probabilities: P(A and B) = P(A) × P(B).
P(A or B) = P(A) + P(B) − P(A and B). The overlap is subtracted so it is not counted twice.
1 − (1 − p)^n. With p = 0.5 and 10 tries, it is 99.9%.

Disclaimer: results are calculated in your browser from the numbers you enter. Check important results before you rely on them for money, health, safety or legal decisions.