Using a probability calculator enables you to calculate the likelihood between different events for the given values. It simplifies complex probability problems and makes it convenient to estimate outcomes for various events, without requiring extensive mathematical knowledge.

## What is Probability?

Probability is a measure of the uncertainty or randomness of an event. It's like a number between (0-1), 0% means (impossible), and 100% means (guaranteed). That tells you how often you expect something to happen if you repeat it many times under the same condition.

This calculation enables you to understand how to find the __expected value__ between 0 and 1. A higher probability shows a higher certainty that the event will happen.

## Probability Formula:

The probability formula is given as:

$$ \text{P(A)}\;=\frac{\text{n(E)}}{\text{n(S)}} $$

Where:

**P(A)** = Probability of the event

**n(E)** = Represent the favorable outcome

**n(S)** = Total number of event

### Probability Formula for Two Events:

\(P(A \text{ and } B) = P(A) \times P(B)\)

## Rules of Probability:

Here are fundamental rules that guide how we __calculate probabilities__ and understand the relationships between various outcomes.

### Rule of Addition:

P(A∪B) = P(A) + P(B) – P(A∩B)

Probability of either event A or event B occurring is the sum of their individual probabilities minus the probability of both happening together.

### Rule of Complementary Events:

P(A’) + P(A) = 1

Probability of an event A happening plus the probability of the opposite event (not A) is always equal to 1.

### Disjoint Events:

P(A∩B) = 0

If events A and B cannot occur simultaneously, they are disjoint (or mutually exclusive), meaning the probability of both events occurring at the same time is zero.

### Independent Events:

P(A∩B) = P(A) ⋅ P(B)

If events A and B happening or not happening do not affect each other, the probability of both events occurring is the product of their individual probabilities.

### Conditional Probability:

P(A | B) = P(A∩B) / P(B)

Probability of event A happening given that event B has already occurred is the probability of both A and B occurring divided by the probability of B.

### Bayes Formula (Bayes' Theorem):

P(A | B) = P(B | A) ⋅ P(A) / P(B)

The Bayes Theorem states the events and the random variables separately.

## How do we find the probability of events?

Finding probability involves a few simple steps. Take a look at each step with the example:

### Example:

let's say we are trying to find the probability of rolling a 5 on a fair six-sided die.

In the probability formula,

P(A) represents the probability of the event A, n(E) is the number of successful outcomes, and n(S) is the total number of possible outcomes.

For rolling a 5 on a fair six-sided die:

- n(E) (number of successful outcomes) = 1 (because there is only one face with a 5)
- n(S) (total number of possible outcomes) = 6 (because there are six faces on the die)

Now, using the formula:

\(P(A) = \frac{n(E)}{n(S)}\)

Put the values into the equation:

\(P(A) = \frac{1}{6}\)

So, the probability of rolling a 5 on a fair six-sided die is \(\frac{1}{6}\), which means for every six rolls, you would expect to get a 5 once on average. You can also verify these results from our probability calculator.

### Find Probability for Two Events:

Let's consider a situation where we are flipping a coin and rolling a die. We want to find the probability of getting heads on the coin flip and rolling an even number on the die.

For this scenario, we have two events:

**Event A:**Getting heads on the coin flip**Event B:**Rolling an even number on the die

For both A and B events that occur together, we use the following formulas:

\(P(A \text{ and } B) = P(A) \times P(B)\)

Let's say:

P(A) (probability of getting heads) = \(\frac{1}{2}\) because there are two equally possible outcomes (heads or tails) when flipping a coin.

P(B) (probability of rolling an even number) = \(\frac{1}{2}\) because there are three even numbers (2, 4, 6) out of the six possible outcomes when rolling a six-sided die.

Now, apply the formula to find the joint probability for events:

\(P(A \text{ and } B) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\)

So, the probability of getting heads on the coin flip and rolling an even number on the die at the same time is \(\frac{1}{4}\).

This means that out of every four times you perform both actions together, you would expect the desired outcome (heads on the coin and an even number on the die) to happen once, on average.

Thankfully, you come to know how to determine the probability of the events occurring at once. You can use the advanced mode given in the probability calculator to calculate the probability for two events.

## How to Use the Probability Calculator?

- Choose probability-finding options from the given drop-down
- Add the statistical values for your events into the given tool section
- Click calculate
- This probability calculator provides you with the likelihood of the occurrence of your chosen events.