Probability Calculator for Two Independent Events
Find P(A and B), P(A or B), P(not A), P(neither) and P(exactly one) for two independent events, from decimals or percentages, free in your browser.
Enter 0 to 1 or a percentage such as 25%. The two events are assumed to be independent.
- P(A and B) = P(A) × P(B)
- 0.2 (20%)
- P(A or B) = P(A) + P(B) − P(A)P(B)
- 0.7 (70%)
- P(not A) = 1 − P(A)
- 0.5 (50%)
- P(not B) = 1 − P(B)
- 0.6 (60%)
- P(neither) = (1 − P(A))(1 − P(B))
- 0.3 (30%)
- P(exactly one) = P(A) + P(B) − 2P(A)P(B)
- 0.5 (50%)
Probability Calculator
This page opens in probability mode. Enter the probabilities of two events, A and B, as decimals from 0 to 1 or as percentages such as 25%, and the calculator gives the chance that both happen, that at least one happens, that each one does not happen, that neither happens and that exactly one happens. Results show as a decimal with the percentage beside it. The page starts with P(A) = 0.5 and P(B) = 0.4.
Formulas for independent events
The calculator assumes the two events are independent: one happening does not change the chance of the other, like two separate coin flips or dice rolls. For dependent events, such as drawing two cards without putting the first back, P(A and B) needs the conditional probability of B given A instead.
How to use it
Steps
- Type Probability of A and Probability of B, for example 0.5 or 50%.
- Read P(A and B) and P(A or B) at the top, and the other results below.
Worked examples
P(A) = 0.5, P(B) = 0.4
P(A and B) = 0.5 × 0.4 = 0.2. P(A or B) = 0.5 + 0.4 − 0.2 = 0.7. P(neither) = 0.5 × 0.6 = 0.3 and P(exactly one) = 0.7 − 0.2 = 0.5.
Percentages: 10% and 0.2
P(A and B) = 0.1 × 0.2 = 0.02, which is 2%, and P(A or B) = 0.1 + 0.2 − 0.02 = 0.28, which is 28%.
Checks
A probability must lie from 0 to 1, or from 0% to 100%. Values such as 1.2 or 150% are refused with a message next to the field. Results print with 12 significant digits, so floating point noise such as 0.020000000000000004 shows as 0.02.
Reading the results
P(A or B) is the chance that at least one event happens, including both. P(exactly one) leaves out the case where both happen, so it is P(A or B) minus P(A and B). P(neither) is the complement of P(A or B): the two always add up to 1. For the example, 0.7 + 0.3 = 1, which is a quick check that the numbers make sense.
Examples of independent events
Rolling a six on one die and a six on another: 1/6 × 1/6 = 1/36, about 0.0278. Two flips of a fair coin both landing heads: 0.5 × 0.5 = 0.25. Rain on two days that are far apart is close to independent, but rain on two days in a row is not, because weather comes in spells.
Privacy
Nothing is uploaded; every calculation runs in your browser and never leaves your device.
Related tools
To count the outcomes behind a probability, such as the number of possible poker hands, use the Combination Calculator or the full Permutation and Combination Calculator. The Percentage Calculator converts between fractions and percentages.
Frequently asked questions
How do I find the probability of A and B?
For independent events, multiply: P(A and B) = P(A) × P(B). With P(A) = 0.5 and P(B) = 0.4, P(A and B) = 0.2.
How do I find the probability of A or B?
Add the two and subtract the overlap: P(A or B) = P(A) + P(B) − P(A and B). For 0.5 and 0.4 that is 0.5 + 0.4 − 0.2 = 0.7.
What does independent mean?
Two events are independent when one happening does not change the chance of the other, like two coin flips. The calculator assumes independence; for dependent events P(A and B) = P(A) × P(B given A).
Can I enter percentages?
Yes. Type 10% or 0.1; both mean the same. Results show as a decimal with the percentage in brackets, such as 0.02 (2%). Values must lie from 0 to 1, or 0% to 100%.
What are P(neither) and P(exactly one)?
P(neither) = (1 − P(A))(1 − P(B)) is the chance that both fail. P(exactly one) = P(A) + P(B) − 2P(A)P(B) is the chance that one happens but not both. For 0.5 and 0.4 they are 0.3 and 0.5.