Two independent events
Input
P(A) = 0.5, P(B) = 0.3
Expected result
P(A and B) = 0.15, P(A or B) = 0.65, P(neither) = 0.35, P(exactly one) = 0.5
Calculators
Probability Calculator
Calculate probabilities for two independent events, conditional probability P(A|B) and binomial outcomes, with every formula shown.
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About this tool
The Probability Calculator answers three common kinds of probability question and shows the formula behind every number. In Two events mode, enter P(A) and P(B) for independent events to get P(A and B), P(A or B), P(not A), P(not B), P(neither) and P(exactly one). Conditional mode returns P(A|B) from P(A and B) and P(B); add P(A) to also get P(B|A), P(A or B) and an independence check. Binomial mode takes n trials, k successes and the success probability p, and returns P(X = k), P(X ≤ k), P(X ≥ k), P(X < k) and P(X > k) with the mean and standard deviation. Probabilities can be typed as decimals, percentages or fractions. Binomial sums are computed with logarithms, so up to 10,000 trials stay accurate and tiny results appear in scientific notation instead of rounding to 0. Two events mode assumes independence; for dependent events, use conditional mode. Everything runs in your browser.
Why use it
Covers independent events, conditional probability and the binomial distribution on one page.
Accepts decimals, percentages and fractions, so 1/6 for a die roll works as typed.
Shows every formula with your numbers substituted, which helps with homework and exam revision.
Gives all five binomial results at once: exactly, at most, at least, fewer than and more than k.
Reports tiny probabilities, such as 1 × 10^-1000, in scientific notation instead of rounding them to zero.
Rejects impossible inputs, such as P(A and B) larger than P(B), with a plain explanation.
How to use
- Pick Two events, Conditional or Binomial at the top.
- Type probabilities as a decimal (0.25), a percentage (25%) or a fraction (1/4).
- For a binomial question, enter the number of trials n, the number of successes k and the chance of success on each trial p.
- Read the results in the cards; each also shows a percentage unless it is extremely small or extremely close to 1.
- Check the worked formulas below the cards to see how every number was calculated.
When it helps
- Statistics homework that combines events with and, or and not.
- Finding the chance of k successes in n tries, such as defective parts in a batch or free throws made.
- Checking whether two events are independent from survey or experiment data.
- Working out odds in dice, card and coin games.
- Sanity-checking conversion counts or pass rates with the binomial distribution.
Examples
Conditional probability
Input
P(A and B) = 0.12, P(B) = 0.4, P(A) = 0.3
Expected result
P(A|B) = 0.3, P(B|A) = 0.4, P(A or B) = 0.58; A and B are independent
3 heads in 10 coin flips
Input
n = 10, k = 3, p = 0.5
Expected result
P(X = 3) = 0.1171875, P(X ≤ 3) = 0.171875, P(X ≥ 3) = 0.9453125; mean 5, SD 1.58114
Tips
- When events can overlap, use P(A or B) = P(A) + P(B) − P(A and B); adding P(A) and P(B) alone counts the overlap twice.
- At least one problems are easiest through the complement: P(at least one) = 1 − P(none).
- For dice, type 1/6 rather than 0.17 so early rounding does not creep into the answer.
- P(X ≥ k) includes k itself and P(X > k) does not; match the one to the wording of your question.
- When a result is extremely close to 1, it is written as 1 minus a tiny number in scientific notation so the remainder is not rounded away.
Frequently Asked Questions
How do you calculate 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) = 0.3, that is 0.15, or 15%.
How do you calculate the probability of A or B?
Use P(A or B) = P(A) + P(B) − P(A and B). For independent events with probabilities 0.5 and 0.3: 0.5 + 0.3 − 0.15 = 0.65.
What is conditional probability?
P(A|B) is the probability of A given that B happened: P(A and B) ÷ P(B). If P(A and B) = 0.12 and P(B) = 0.4, then P(A|B) = 0.3.
How do I know if two events are independent?
A and B are independent when P(A and B) = P(A) × P(B), which is the same as P(A|B) = P(A). Enter P(A) in conditional mode and the calculator checks this for you.
What is the binomial probability formula?
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), where C(n, k) counts the ways to choose which k of the n trials succeed. For 3 heads in 10 fair coin flips: 120 × 0.5^10 = 0.1171875.
What is the difference between P(X ≤ k) and P(X < k)?
P(X ≤ k) includes exactly k successes, while P(X < k) stops at k − 1. For 10 coin flips, P(X ≤ 3) = 0.171875 but P(X < 3) = 0.0546875.
Can I enter percentages or fractions?
Yes. Type 25% for a percentage or 1/6 for a fraction. A number above 1 without a % sign is rejected, so 25 is never silently read as 25%.
How large can n be in binomial mode?
Up to 10,000 trials. The calculator works with logarithms, so very small results stay accurate; for example, 1,000 successes in 1,000 trials at p = 0.1 shows 1 × 10^-1000.
What do the mean and standard deviation tell me?
The mean, n × p, is the expected number of successes, and the standard deviation, √(n × p × (1 − p)), is the typical spread around it. For 10 fair coin flips, the mean is 5 and the standard deviation is about 1.58.
Explore the category
Glossary
- Independent events
- Events where one happening does not change the probability of the other.
- Conditional probability
- The probability of an event given that another event has occurred, written P(A|B).
- Complement
- The event that A does not happen; P(not A) = 1 − P(A).
- Binomial distribution
- The distribution of the number of successes in n independent trials that each succeed with the same probability p.
- Binomial coefficient
- C(n, k), the number of ways to choose k items from n; C(10, 3) = 120.
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