Probability and Normal Distribution Calculator
Calculate the probability of a single event, the union or conditional probability of two events, the intersection of two independent events, a binomial result for up to 1,000 trials, or a normal distribution from a mean and a standard deviation. Enter probabilities as decimals from 0 to 1, so 25% is 0.25. Results show six decimals. A chance below 0.000001 prints in e-notation, so 1 in 292,201,338 shows 3.422e-9.
How to use the Probability Calculator
- Enter your numbers in the Probability Calculator above.
- Press the Calculate button to see the result.
- To try other numbers, change any value and press the button again.
Key probability formulas
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) · · · P(A | B) = P(A ∩ B) / P(B) · · · For independent events: P(A ∩ B) = P(A) × P(B)
Union is "or" (either or both); intersection is "and" (both). Conditional probability is the probability of A given that B has occurred. Independence means knowing B tells you nothing new about A. In the normal distribution mode, z = (x − μ) ÷ σ turns x into standard deviations from the mean, and the probability is the area under the bell curve up to z.
Worked example
Draw one card from a 52-card deck. Basic mode gives P(Heart) = 13 / 52 = 0.25, P(King) = 4 / 52 = 0.076923 and P(King of Hearts) = 1 / 52 = 0.019231. In union mode, 0.25 + 0.076923 − 0.019231 = 0.307692, which subtracts the King of Hearts counted twice. In conditional mode, P(King given Heart) = 0.019231 / 0.25 = 0.076924. The exact value is 1/13 = 0.076923. The last digit differs because the input 0.019231 is itself rounded. For a normal distribution with a mean of 100 and a standard deviation of 15, P(X ≤ 130): z = (130 − 100) ÷ 15 = 2, so P(X ≤ 130) = 0.97725 and P(X ≥ 130) = 0.02275.
Frequently asked questions
What is conditional probability?
The probability of one event given that another has occurred. P(A|B) reads "probability of A given B." It changes the sample space to only the outcomes where B happens.
How do I know if two events are independent?
A and B are independent when P(A∩B) = P(A) × P(B), which means knowing B does not change the probability of A. Hearts and kings pass that test: 0.25 × 0.076923 = 0.019231, the chance of the King of Hearts. The independent mode takes independence as given and does not test it.
Why does it refuse some values?
Because they cannot belong to the same two events. P(A∩B) cannot be larger than P(A) or P(B), and a union cannot pass 1. P(A) = 0.9, P(B) = 0.8 and P(A∩B) = 0.5 would give a union of 1.2, so you get a message and no result. A percentage typed as 25 gets a message too.
What does the binomial mode calculate?
The chance of exactly k successes in n independent trials that each succeed with probability p, and the chance of k or fewer. For 3 heads in 10 fair coin flips, enter n = 10, k = 3 and p = 0.5. The calculator returns C(10,3) = 120, P(X = 3) = 0.117188 and P(X ≤ 3) = 0.171875. It takes up to 1,000 trials and keeps every digit while it works, so 0 heads in 1,000 flips shows 9.333e-302.
How do I find a normal distribution probability?
Pick Normal Distribution, enter the mean and the standard deviation, then choose P(X ≤ x), P(X ≥ x) or P(a ≤ X ≤ b). With a mean of 100 and a standard deviation of 15, P(X ≤ 130) = 0.97725, because 130 is z = 2 standard deviations above the mean. P(85 ≤ X ≤ 115) = 0.682689, the familiar 68% within one standard deviation. A mean of 0 and a standard deviation of 1 is the standard normal curve.
How do I find the value for a given probability?
Choose "x from a probability" and enter the share below x as a decimal. With a mean of 100 and a standard deviation of 15, a probability of 0.95 gives x = 100 + 1.644854 × 15 = 124.672804, so 5% of values lie above about 124.67. For the z-score alone, the z-score calculator takes a value, a mean and a standard deviation.