# Probability of 3 Events Calculator

Created by Davide Borchia
Last updated: Nov 30, 2022

Toss a coin, roll a dice, and ask someone to tell you a number: we can tell you the chance of you guessing the outcomes of these events with our probability of 3 events calculator.

In this calculator, you will learn:

• What is an independent event;
• The most important rules of probability: calculation of the probability of multiple events;
• Which are the possible combinations of the probability of 3 events;
• The formulas for the probabilities of 3 events (3 events, exactly one and two events, at least one and two events, and no events).

Don't take any chance on your homework, and check your results with our tool!

## What are independent events?

In probability, independent events are entirely disjointed events: the probability of one occurring (or not) does not influence the others. Imagine two people in different cities throwing a coin: the outcome of each toss is entirely independent of the other. On the other hand, the first full moon of spring and the date of Easter are dependent events as you can learn from the Easter date calculation formula - we can predict one based on the other.

Independent events have an important role in statistics and probability since dealing with them is way easier than dealing with events that have some degree of correlation.

## How to calculate the probability of multiple events?

There are simple rules to calculate the probability of multiple events. Assume that the events $A$, $B$, etc.. are independent, and define:

• $P(A)$ the probability of $A$ occurring; and
• $P(B)$ the probability of $B$ occurring,
• ...

We can already define the probability of an event not happening. We use the complement rule, which states that given an event $A$ with occurrence probability $P(A)$, the probability of not occurrence is:

$\footnotesize P(\bar{A})=1-P(A)$

Now we can introduce the rules to calculate the probability of occurrence of one, none, or multiple events.

• To calculate the probability of a given number of events happening together, we can multiply the probabilities of occurrence of the disjointed events.
• To calculate the probability of at least some events happening out of the total, we need to sum the correct probabilities;

## Take three: the probability of 3 independent events

In the probability of 3 events calculator, we will deal with just three independent events. It will give us enough material to calculate many combinations. But when do we need to know the probability of 3 independent events?

It may be useful while playing a game or calculating the likelihood of bad weather during a trip. Even if it doesn't always look like it, the probability is useful, and you can find it everywhere!

Let's see it from a practical point of view of three parallel resistors. What is the probability that at least two of them will be still working after ten years if the probability that a resistor breaks after ten years is 20%? Using our calculator, you can immediately see the result is nearly 90 % if you set $P(A) = P(B) = P(C) = 80 \text{ } \%$. It's quite handy, isn't it? Check Omni's if you want to know more about this topic.

## How to calculate the probability of 3 events

Here we will explain how to calculate the probability of multiple events. With 3 events, we have various choices:

• Calculate the probability of all 3 events occurring;
• Calculate the probability of at least one occurring;
• Calculate the probability of exactly one occurring;
• Calculate the probability of at least two occurring;
• Calculate the probability of exactly two occurring; and
• Calculate the probability of none of them occurring.

The formulas for the probability of 3 events are closely related to the formulas we've seen for two events. Introducing the event $C$ with probability $P(C)$, let's discover the possible combinations!

#### The probability of all 3 events occurring

To calculate the probability of all 3 events occurring, we multiplicate the disjointed probabilities:

$\footnotesize P(A\cap B\cap C)= P(A)\cdot P(B)\cdot P(C)$

It's easier to see the meaning of this calculation on an Euler-Venn diagram:

#### The probability of at least one event occurring

When considering 3 events, the formula for the probability of at least one of them occurring asks us to sum the probabilities of the three events happening disjointedly and removing the redundant products:

\footnotesize \begin{align*} P&(A\cup B\cup C) \!=\! P(A)+P(B)+P(C)\\ -&P(B\cap C)-P(A\cap C)-P(A\cap B)\\ +&P(A\cap B\cap C) \end{align*}

Take a look at the Euler-Venn diagram again: we highlighted the area corresponding to this quantity!

#### The probability of exactly one event occurring

If you want to calculate the probability of *exactly one event happening out of three, you need to exclude the other two. We have three combinations:

• $A$ happens, and $B$ and $C$ don't: $P(\bar{A}\cap \bar{B}\cap \bar{C})$;
• $B$ happens, and $A$ and $C$ don't: $P(B\cap \bar{A}\cap \bar{C})$; and
• $C$ happens, and $A$ and $B$ don't: $P(C\cap \bar{A}\cap \bar{B})$;

Sum these probabilities to get the final result:

\footnotesize \begin{align*} P&(A\cap \bar{B}\cap \bar{C})+P(B\cap \bar{A}\cap \bar{C})+\\ P&(C\cap \bar{A}\cap \bar{B})\\ =&(P(A)\cdot P(\bar{B})\cdot P(\bar{C}) )\\ +&(P(B)\cdot P(\bar{A})\cdot P(\bar{C}))\\ +&(P(C)\cdot P(\bar{A})\cdot P(\bar{B})) \end{align*}

Applying the rule of the complement, and rearranging the formula, we can find the following result:

\footnotesize \begin{align*} P&(A\cap \bar{B}\cap \bar{C})+P(B\cap \bar{A}\cap \bar{C})+\\ P&(C\cap \bar{A}\cap \bar{B})\\ =&P(A\cup B\cup C)- P(C\cap B)\\ -&P(A\cap C)- P(A\cap B) \\ +& 2\cdot P(A\cap B\cap C) \end{align*}

#### The probability of at least two and exactly two events occurring

For the probabilities of these combinations of independent events, we calculate two equations similar to the previous ones:

• To calculate the probability of at least two events, we consider the pairs $P(A\cup B)$, $P(A\cup C)$, and $P(B\cup C)$. You can try your hand with the math, starting from the Euler-Venn diagram.
• To calculate the probability of exactly two events, we need to consider only the intersections between two sets.

#### The probability of no events occurring

For the probability of no events occurring out of three independent events, we calculate the following expression:

\footnotesize \begin{align*} P(\emptyset) = 1 - P(A\cup B\cup C) \end{align*}

Where the second term corresponds to the probability of at least one event occuring.

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