Probability Calculator

This probability calculator combines two events, A and B. It finds the probability that both happen, that at least one happens, that neither happens, and the conditional probability of A given B. It handles independent events, mutually exclusive events and events where you know the overlap, using exact fractions throughout.

Use decimals or fractions, e.g. 1/6.
Try:

How to use the probability calculator

  1. Enter P(A) and P(B) as decimals or fractions.
  2. Choose how the events are related.
  3. Press Calculate to see every combined probability.

Formula

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)
P(A∣B)=P(A∩B)P(B)P(A \mid B) = \frac{P(A \cap B)}{P(B)}

Worked example: Coin heads and die 6 (independent)

  • P(A): 1/2
  • P(B): 1/6
  • A and B are: Independent

✓ Answer checked

P(A∩B)=112=0.083333…,P(A∪B)=712=0.583333…P(A \cap B) = \frac{1}{12} = 0.083333\ldots,\quad P(A \cup B) = \frac{7}{12} = 0.583333\ldots
P(not A)
12=0.5\frac{1}{2} = 0.5
P(not B)
56=0.833333…\frac{5}{6} = 0.833333\ldots
P(A given B)
12=0.5\frac{1}{2} = 0.5
P(neither)
512=0.416667…\frac{5}{12} = 0.416667\ldots
P(exactly one)
12=0.5\frac{1}{2} = 0.5
00.10.20.30.4A only: 0.4166666666666667A onlyboth: 0.08333333333333333bothB only: 0.08333333333333333B onlyneither: 0.4166666666666667neither
The four outcomes: these probabilities always add up to 1.
Step-by-step working (3 steps)

  1. P(A and B) P(A∩B) = P(A)P(B)

    For independent events, multiply.

    12×16=112\frac{1}{2} \times \frac{1}{6} = \frac{1}{12}

  2. P(A or B) Addition rule

    Add, then subtract the overlap so it is not counted twice.

    12+16−112=712\frac{1}{2} + \frac{1}{6} - \frac{1}{12} = \frac{7}{12}

  3. Complements

    P(not A) = 1 − P(A).

    1−12=121 - \frac{1}{2} = \frac{1}{2}
Formulas used
P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)
P(A∣B)=P(A∩B)P(B)P(A \mid B) = \frac{P(A \cap B)}{P(B)}
How this was checked
  • ✓ The four separate outcomes add up to exactly 1.

Frequently asked questions

What is the probability of a coin landing heads and a die showing 6?

They are independent, so multiply: ½ × ⅙ = 1/12.

What does mutually exclusive mean?

The events cannot happen together, so P(A and B) = 0 and P(A or B) = P(A) + P(B).

Why subtract P(A and B) in the 'or' rule?

Outcomes in both A and B would otherwise be counted twice.

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