# Probability Calculator

Created by Krishna Nelaturu
Last updated: Jun 26, 2022

Welcome to our probability calculator, where you can determine the chance of different types of outcomes possible based on the probabilities of two independent events. You can also find an event's probability when you repeat the trial multiple times.

If you find this affair of calculating the probabilities of two events confounding, scroll down further because we're going to break this concept down and answer some fundamental questions:

• What is the probability definition?
• What are the different likely outcomes based on two events?
• Using the probability formula, how do you find the probabilities of different outcomes based on two independent events?
• How does repeating the trial affect an event's probability?

## Probability definition: What is probability?

Suppose it's your turn to roll the dice in your favorite board game, and you win if you roll a four or a six. How do you determine your odds of victory? A game of chance (like a dice game) where the outcome of a trial (rolling the dice) is random is a perfect setting to understand probability which is opposed to, e.g., for the mechanical advantage that is known to be 100 % correct in every case.

In mathematical terms, we define probability as the ratio of the number of favorable outcomes to the total number of possible outcomes. We can express it using the probability formula:

$\small P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

Here $P(A)$ is the probability of the event $A$. You can see that the value of any event's probability must lie within $0-1$. Sometimes it is convenient to speak about it in percentages.

In the dice example above, you win if you roll a four or a six, meaning you have two favorable outcomes out of six possible outcomes. Hence, your probability of victory is $\frac{2}{6} = \frac{1}{3}$.

Before we move to the next section, let's establish the following terms:

• A trial is an experiment or a process that leads to random outcomes, like rolling a dice or flipping a coin.
• An event is an outcome of a trial that we have a specific interest in, like getting heads when you flip a coin.
• Two events are independent if the occurrence of one event doesn't affect the probability of the other, like getting two heads when you flip two coins.

An example of probability in physics is radioactive decay, which we describe using the half life calculator to see how quickly unstable material reduces its mass.

## How do you find the probability of different outcomes based on two events?

Consider the following independent events when you roll a dice:

• $A$ is the event when you roll an even number. Since we have three even numbers $(2,4,6)$ on a dice, $P(A) = \frac{3}{6} = \frac{1}{2}$.
• $B$ is the event when you roll a prime number. Since a dice has three primes $(2,3,5)$, $P(B) = \frac{3}{6} = \frac{1}{2}$.

How do you find the probability of both A and B occurring together? Can we calculate the probability of at least one event occurring? Is it possible to calculate the probability of A and B not occurring? In the following table, we explore such different combinations of these two independent events and their probability formulae.

Event combo

Probability formula

$\small P(A \cap B)$ $A$ AND $B$

$\small P(A) * P(B)$

$\small P(A \cup B)$ $A$ OR $B$

$\small P(A) + P(B) - P(A \cap B)$

$\small P(A \triangle B)$ $A$ XOR $B$

$\small P(A) *P(B') + P(B) * P(A')$

$\small P((A \cup B)')$ neither $A$ nor$B$

$\small P(A') * P(B')$

$\small P(A')$ not $A$

$\small 1- P(A)$

$\small P(B')$not $B$

$\small 1- P(B)$

Where:

• $\small P(A \cap B)$ is the probability of $A$ and $B$ occurring together;
• $\small P(A \cup B)$ is the probability of either $A$ or $B$;
• $\small P(A \triangle B)$ is the probability of exactly one of the two events occurring;
• $\small P((A \cup B)')$ is the probability of neither $A$ nor $B$ occurring;
• $\small P(A')$ is the probability of $A$ not occurring; and
• $\small P(B')$ is the probability of $B$ not occurring.

Using these probability definitions and formulae, find answers to our earlier questions. Check your results using this probability calculator. Wonder how to extend this to include three events? Learn more with our probability of three events calculator.

## How does repeating the trial affect an event's probability?

When we repeat a trial multiple times, say rolling a dice multiple times, the probability of the events changes based on the number of repetitions $n$. For an event $A$:

\scriptsize \begin{align*} P(A \text{ always occuring}) &= P(A)^n\\ P(A \text{ never occuring}) &= P(A')^n\\ P(A \text{ occuring at least once}) &= 1-P(A')^n\\ \end{align*}

Suppose you want to calculate the probability of at least one $6$ out of three successive dice rolls.

1. First, you determine the probability of getting a $6$ in one roll, $P(6) = \frac{1}{6}$.
2. Next, find $P(6') = 1- \frac{1}{6} = \frac{5}{6}$.
3. Finally, use the probability formula above to get:
\qquad \scriptsize \begin{align*} P(6 \text{ at least once}) &= 1 - P(6')^3\\ &= 1 - \left(\frac{5}{6}\right)^3\\ &= 1 - \left(\frac{125}{216}\right)\\ P(6 \text{ at least once})&= \frac{91}{216} = 42.13\% \end{align*}

## How to use this probability calculator of two events

Our probability calculator of two events is perfect for anyone who wishes to calculate the probabilities of A and B and the likelihood of their different combinations.

To determine the probability of the different combinations of two events in a trial, follow these steps:

1. Enter the probabilities of events A and B.
2. Under the "Which probability do you want to see?" section, choose which combination of these two events is of interest to you. You can also opt to see all of them.
3. Sit back and relax. The calculator will provide the answer you want instantly.

To find out how likely an event is when we repeat the trial multiple times, follow these steps:

1. Enter the probability of A or B. You can enter both if you wish to compare.
2. Under the "Probabilities for a series of events" section, enter the number of trial repetitions in the When trying field. Also, choose which type of event interests you.
3. The calculator will show you how the repetition has changed the chances of the event.

If you don't want to rely on probability during your trips, our gas cost calculator is a perfect tool to plan it effectively.

Krishna Nelaturu
Probabilities of single events
Probability of A, P(A)
%
Probability of B, P(B)
%
Which probability do you want to see?
A and B both occurring P(A∩B)
%
Probabilities for a series of events
When trying
times...
... A always occurring
Probability
%
If you want to find the conditional probability, check our Bayes' Theorem Calculator!
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