Probability Calculator

What is a probability calculator?

A probability calculator works out how likely combinations of two events are once you know the chance of each one on its own. You enter the probability of event $A$ and the probability of event $B$ as percentages, and the calculator returns four combined probabilities: both events together, at least one of them, neither of them, and $A$ happening while $B$ does not.

This tool assumes the two events are independent — the outcome of one has no effect on the outcome of the other. Rolling a die and tossing a coin, or two separate machines each having a fixed failure rate, are classic examples of independent events.

How does the calculator work?

You supply two inputs, each between 0% and 100%:

• P(A) — the probability that event $A$ occurs.
• P(B) — the probability that event $B$ occurs.

Because the events are independent, the joint probabilities follow directly from multiplication. Working in percentages, every product is divided by 100 to keep the result on a 0–100% scale. The calculator then reports:

• P(A and B) — both events occur.
• P(A or B) — at least one of the two events occurs.
• P(neither A nor B) — neither event occurs.
• P(A but not B) — $A$ occurs while $B$ does not.

Formula

For two independent events with probabilities $p_A$ and $p_B$ (written as decimals):

When the inputs are entered as percentages, each product term is divided by 100. For example $P(A \cap B) = \dfrac{P(A) \cdot P(B)}{100}$ with $P(A)$ and $P(B)$ in percent.

Worked examples

1. Two fair coins, P(A) = P(B) = 50%. Both heads: $50 \times 50 / 100 = 25\%$. At least one head: $50 + 50 - 25 = 75\%$. Neither head: $50 \times 50 / 100 = 25\%$. First heads but not the second: $50 \times 50 / 100 = 25\%$.

2. P(A) = 20%, P(B) = 30%. Both: $20 \times 30 / 100 = 6\%$. Either: $20 + 30 - 6 = 44\%$. Neither: $80 \times 70 / 100 = 56\%$. A but not B: $20 \times 70 / 100 = 14\%$.

Notes

• The four results are related: $P(A \cup B)$ and $P(\text{neither})$ always add up to 100%, because “at least one” and “none” are complementary outcomes.
• Independence is the key assumption. If knowing that $A$ happened changes the chance of $B$, the events are dependent and you need conditional probability instead — see the Bayes’ theorem calculator.
• To combine the same event over many repeated trials (such as several coin flips in a row), use the coin flip probability calculator, which applies the binomial distribution.

FAQ

Do the probabilities have to add up to 100%? No. $P(A)$ and $P(B)$ are independent inputs and each can be anything from 0% to 100%. They describe two separate events, not two outcomes of one event.

What does “independent” mean here? Two events are independent when the occurrence of one does not change the probability of the other. Only under independence does $P(A \cap B) = P(A) \cdot P(B)$ hold.

How do I handle mutually exclusive events? If two events cannot both happen, they are not independent, and $P(A \cap B) = 0$. This calculator is designed for independent events, so it is not the right tool for mutually exclusive ones.