Joint Probability Calculator
Find the chance that two independent events both occur and display their joint probability as both a fraction and a percentage.
How the result changes
Only probability of event a changes; all other inputs stay at your current values. The line illustrates the formula across the shown range. This is not a forecast of real-world outcomes.
About this calculation
Joint Probability Calculator: Find the chance that two independent events both occur and display their joint probability as both a fraction and a percentage. Enter probability of event a, probability of event b; the reported quantities are joint probability, joint probability (%). The equations and examples below show how those inputs lead to the result and where the model's assumptions matter.
Formula
P(A and B) = P(A) × P(B), if independent
Worked example
With Probability of event A: 0.4, Probability of event B: 0.3, joint probability is 0.12.
Method and inputs
When events A and B are independent, learning that A occurred does not change the chance of B. Multiply their individual probabilities to obtain the chance that both occur. The calculator shows both a decimal probability and percentage.
Worked example
If A has probability 0.4 and B has probability 0.3, independence gives 0.4 × 0.3 = 0.12, or 12 percent, for their intersection. This is lower than either individual chance in this example.
Two independent coin flips both landing heads have probability 0.5 × 0.5 = 0.25. The chance of at least one head is 0.75, a different event. Drawing two cards without replacement is not analogous because the first draw changes the second probability.
How to read the result
The result is the intersection of two events, not the chance of at least one. For independent events, the latter is pA + pB - pA×pB. Both input probabilities must represent the same experiment or coherent sequence.
If measured data show P(B given A) = 0.6 while P(B) = 0.3, the events are dependent. Multiplying P(A) by 0.3 would understate their intersection; use the conditional 0.6 instead. The independence claim should come from the situation, not convenience.
Assumptions and checks
Independence is a substantive assumption. For dependent events, the correct product is P(A) × P(B given A), and using the unconditional probability of B may be wrong. Probabilities must lie between zero and one.
Do not enter 40 for a 40-percent probability; enter 0.4. A pair of events can be mutually exclusive without being independent unless one has zero probability. Carefully define each event before multiplying.
The decimal and percentage outputs should always represent the same chance: multiply the decimal by one hundred. If either input is zero, their joint probability is zero; if both inputs are one, the intersection is certain under the independence model.
Common questions
What does the joint probability calculator calculate?
It reports the chance of both independent events happening, in decimal and percent form.
When should I recalculate or check another method?
Recalculate using a conditional probability model when events share causes, are drawn without replacement or otherwise influence one another. Empirical data can reveal dependence that simple labels hide.
Inputs
- Probability of event A
- Probability of event B