4 Event Probability Calculator: All, None, At Least One

Probability of exactly k events
k events occur	Probability	At least k
0	0.126	1
1	0.383	0.874
2	0.365	0.491
3	0.117	0.126
4	0.009	0.009

All fourProduct of the four

At least one1 − Π(1 − pᵢ)

Exactly kSum over combinations

AssumesIndependence

What Is Probability of 4 Events?

For four independent events, the probability that all occur is the product of the four probabilities and the probability that none occur is the product of the four complements. "At least one" is 1 minus the chance that none occur, and "exactly k" sums over every way of choosing which k events happen.

The calculator applies these rules to your four probabilities and lists the result for every count from zero to four. When all four events have the same probability the problem reduces to a binomial distribution, and the page shows that special case.

Below you will find the formulas, a variables table, a worked example and answers to common questions such as what independence means and how this differs from the three-event version.

How to Use the Probability of 4 Events Calculator

Enter P(A), P(B), P(C) and P(D) as decimals, percentages or fractions.
Read “all”, “none” and “at least one” first.
Use the table for the probability of exactly 0, 1, 2, 3 or 4 events, and the cumulative column for “at least k”.
Check the chart to see which count of events is most likely.

Probability of 4 Events Formula

All four (independent)

P(A\cap B\cap C\cap D)=\prod_{i}p_i

At least one

P(\ge1)=1-\prod_{i}(1-p_i)

Exactly k occur

P(K=k)=\sum_{|S|=k}\prod_{i\in S}p_i\prod_{j\notin S}(1-p_j)

Equal probabilities (binomial special case)

P(K=k)=\binom{4}{k}p^k(1-p)^{4-k}

Probability of 4 Events Variables Explained

Symbol	Meaning	Notes
P(A)…P(D)	Probability of each event	0 to 1
P(all)	All four occur	Product of the four
P(none)	None occur	Product of the complements
P(≥1)	At least one occurs	1 − P(none)
k	Number of events that occur	0 to 4

Probability of 4 Events Worked Example

Using the example values (press “Load example” above to try them):

Assuming independence, P(all) = 0.6 × 0.5 × 0.3 × 0.1 = 0.009.
P(none) = (1 − 0.6) × (1 − 0.5) × (1 − 0.3) × (1 − 0.1) = 0.126.
P(at least one) = 1 − P(none) = 0.874.
P(exactly k) is found by adding the probabilities of every combination with k events happening and the rest failing: k=0: 0.126, k=1: 0.383, k=2: 0.365, k=3: 0.117, k=4: 0.009.

Result: All 4 events occur P(A ∩ B ∩ C ∩ D) = 0.009; None occur = 0.126; At least one occurs P(A ∪ B ∪ …) = 0.874.

Worked example

A project depends on four independent risks occurring with probabilities 0.6, 0.5, 0.3 and 0.1. The chance none occurs is 0.4 × 0.5 × 0.7 × 0.9 = 0.126, so the chance at least one occurs is 0.874. All four occurring has probability 0.6 × 0.5 × 0.3 × 0.1 = 0.009. The most likely number of risks to occur is shown by the tallest bar in the chart.

When all four probabilities are equal

If each event has the same probability p, the number of events that occur follows a binomial distribution with n = 4. For example, with four fair coin flips (p = 0.5) the probabilities of 0 to 4 heads are 1/16, 4/16, 6/16, 4/16 and 1/16. For more than four events, use the binomial calculator.

Four fair coin flips
Heads	Combinations	Probability
0	1	1/16 = 6.25%
1	4	4/16 = 25%
2	6	6/16 = 37.5%
3	4	4/16 = 25%
4	1	1/16 = 6.25%

What if the events are dependent?

If one event changes the probability of another, as when drawing without replacement, independence fails and multiplying marginal probabilities gives a wrong answer. You then need conditional probabilities or a joint probability model. When events are mutually exclusive, the chance that one of them happens is the sum of their probabilities.

Where Probability of 4 Events Is Used

Multi-component reliability

Chance that all four parts work, or that at least one fails.

Sequences of independent outcomes

Combining four separate chances.

Coursework

Checking answers on all, none and exactly-k probabilities.

Risk of at least one event

Estimating the chance that something occurs at least once across four independent chances.

Common Probability of 4 Events Mistakes to Avoid

Probabilities of independent events multiply; those of mutually exclusive events add.
Check that none of your four events is a subset or restatement of another.
The complement is the fastest route to “at least one”.

Probability of 4 Events Terms and Definitions

Poisson-binomial distribution: The distribution of the number of successes among independent trials with different success probabilities.

Probability of 4 Events FAQs

How do I calculate the probability of 4 events?

For independent events multiply the four probabilities to get the chance all occur. For at least one, compute 1 minus the product of (1 − p) for each event.

How do I find the probability of exactly two of four events?

Add the probability of every pair occurring while the other two fail. This calculator does the sum for you.

Does the order of events matter?

No. The results do not depend on which event is called A, B, C or D.

What if all four have the same probability?

Then the number of occurrences follows a binomial distribution with n = 4.

Can the probabilities add up to more than 1?

Yes, for independent events they are separate probabilities and need not sum to 1. Only mutually exclusive, exhaustive events must sum to 1.

What does the chart show?

The probability that exactly 0, 1, 2, 3 or all 4 of the events occur.

4 Event Probability Calculator

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