# Probability of 3 Events Calculator

By Rita Rain
Last updated: Sep 08, 2021

Roll a die, toss a coin, do anything random, and use this probability of 3 events calculator to determine the:

• Probability of at least one event out of three occurring (union of three events);
• Probability of the intersection of three events (all three happening);
• Probability of exactly one event happening; and
• Probability of none of them occurring.

Gain insight into probability rules and decide whether buying a lottery ticket is really worth it.

## Probability calculator for 3 events

This probability calculator works for three independent events. Enter the probability of each event as a percentage, or change the unit to decimals. Once you fill in the three fields, the calculator will output the:

• Probability at least one event occurs out of the three:

`P(A ∪ B ∪ C)`;

• Probability of all three events happening:

`P(A ∩ B ∩ C)`;

• Probability that exactly one of three events happens:

`P(A ∩ B' ∩ C') + P(A' ∩ B ∩ C') + P(A' ∩ B' ∩ C)`;

• Probability that none of the events occur:

`P(∅)`.

Depending on which values you're given, you can also input two numbers into the first section of this calculator and one in the second section.

Remember that each probability has to be a number between 0 and 1 (0% and 100%) inclusive. You can change the unit from percentage to a decimal if you please.

## What are the rules of probability?

Here are the basic rules of probability:

• Probability takes values between 0 (no chance) and 1 (certain) inclusive.

• Complement Rule (probability that an event doesn't occur): `P(A') = 1 - P(A)`.

• Addition rule: `P(A ∪ B) = P(A) + P(B) – P(A ∩ B)`.

• Multiplication rule: `P(A ∩ B) = P(A) × P(B)` for independent events.

`P(A ∩ B) = P(A) × P(B | A) = P(B) × P(A | B)` for dependent events, where `P(B | A)` and `P(A | B)` are the conditional probabilities.

## Probability rules for three events & how to calculate the probability of three independent events

Below you'll find the probability rules used in this probability of 3 events calculator. Use them when you need to calculate the probability of three independent events by hand:

• Multiplication rule - To calculate the probability of the intersection of three independent events, multiply the probabilities of each event together:

`P(A ∩ B ∩ C) = P(A) * P(B) * P(C)`

• Addition rule - To find the probability of the union of three events (probability that at least one event occurs of the three), sum the probabilities of three events and subtract their intersections:

`P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)`

If we apply the multiplication rule we get the formula for the probability of union of three events only in terms of `P(A)`, `P(B)`, and `P(C)`:

`P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A) * P(B) - P(A) * P(C) - P(B) * P(C) + P(A) * P(B) * P(C)`

To calculate the probability of exactly one event happening:

1. We sum the probabilities for three possible scenarios:

1. A happens, B and C don't;
2. B happens, A and C don't; and
3. C happens, A and B don't:

`P(A ∩ B' ∩ C') + P(A' ∩ B ∩ C') + P(A' ∩ B' ∩ C)`.

2. After applying the multiplication rule we get:

`P(A ∩ B' ∩ C') + P(A' ∩ B ∩ C') + P(A' ∩ B' ∩ C) = P(A) * P(B') * P(C') + P(A') * P(B) * P(C') + P(A') * P(B') * P(C')`.

3. Now we'll use the complement rule and we get a formula for which we only need P(A), P(B), and P(C):

`P(A) * P(B') * P(C') + P(A') * P(B) * P(C') + P(A') * P(B') * P(C) = P(A) * (1 - P(B)) * (1 - P(C)) + (1 - P(A)) * P(B) * (1 - P(C)) + (1 - P(A)) * (1 - P(B)) * P(C)`.

To find the chance that none of the three events happen, we use the complement rule for the opposite event, in which at least one event happens (we've already derived the formula for this one):

`P(∅) = 1 - (P(A) + P(B) + P(C) - P(A) * P(B) - P(A) * P(C) - P(B) * P(C) + P(A) * P(B) * P(C))`

## FAQ

### How to find probability of A and B ?

To find the probability of A and B occurring (assuming A and B are independent events, that is, the occurrence of one doesn't influence the occurrence of the other):

1. Determine the probability of A.

2. Determine the probability of B.

3. Multiply the probability of A by the probability of B:

`P(A ∩ B) = P(A) × P(B)`.

### How to find probability of A or B?

To find the probability of A or B (one of them happening but not both):

1. Calculate the probability of A.

2. Find the probability of B.

3. Determine the probability that both A and B will occur by multiplying them.

4. Use the formula:

`P(A ∪ B) = P(A) + P(B) − P(A ∩ B)`,

that is, add the probability of A to the probability of B and subtract the product of the probabilities of A and B.

Note: we assume events A and B are independent.

### What is the probability of an impossible event?

The probability of an impossible event is 0 (or 0%). This is the lowest value probability can have.

### What is the probability of an event that is certain?

The probability of an event that is certain equals 1 (or 100%). This is the largest value probability can have.

### What are independent events in probability?

In probability theory, independent events are events for which the occurrence of one doesn't affect the occurrence of the other. For example, when rolling a die, getting "1" and getting "2" are independent events. Rolling "1" doesn't affect whether you'll get "2" (or any other number) in the next roll.

### How do I calculate probability?

To calculate the probability of event A, divide the number of outcomes favorable to A by the total number of possible outcomes. For example, the chance of getting "5" in a die roll equals 1/6, because we want one case, that is rolling a "5", out of six possible outcomes.

Rita Rain
Probability of events:
P(A)
%
P(B)
%
P(C)
%
Probability of:
All three occurring
%
At least one occurring
%
Exactly one occurring
%
None of them occurring
%
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