How To Find The Probability Of An Event

In mathematics, the probability of events describes the likelihood of an event occurring. In other words, it helps us measure how likely it is that something will happen. Probability is a simple concept and not too difficult to grasp — but if you’ve been learning these concepts with math class in mind, you’ll soon realize that there are many ways to think about probability. One thing that’s important to note about probability is how easily estimations are made in probability. The probability formulas make it easy to determine the probability of events in just a few steps.

History of Probability: 

A Greek mathematician named Nicomachus of Gerasa has attributed the first written work on probability in 300 BC (though there is some evidence that it was used as early as the 6th century BC). He divided possible outcomes into three categories: events with certain outcomes, events with uncertain outcomes, and events without an outcome. Events without an answer were labelled “doubtful.” Studying this area of mathematics was proposed to the Ancient Greeks by Eudoxus.

Concept of Probability: 

Each event has a probability, a number between 0 and 1. This number reflects the likelihood that an event will occur. 

For example, there is a 0.095 chance of winning the lottery by buying all the tickets in one week (9 out of 100)

Probability formulas:

P(A) = 1 – P(not A) or P(A) = 1 – P(not A).

To calculate the probability of an event, let the probabilities of each hypothesis be unknown. Then if the event A is true, then by definition, P(A) = 1. If the event A is not true (not A), then P(A) = 0.

P(not A) = 1- P(A) or P(not A) = 1 – P(A).

The types of probability and their probability formulas:

There are several different types of probability, each with its associated probability formulas for finding probabilities of events. These include:

1. Probability of an event or outcome

The probability of the event is what it’s called when we refer to the event itself. 

For example, the formula for finding the probability of an event is P(A). A six-sided die roll probability is written as 6/36 (1 in 6). This probability is the same as one, but it’s written as a fraction because you have to find the probability of an event.

2. Probability based on some other factors

In this case, the probability is what we are given to find the event. 

For example, if someone says in an interview that there’s a 90% chance they would receive an offer for the job they’re applying for, we might write P(X | Y).

3. Probability based on a random number generator

It is the type of probability that computer programmers and mathematicians most often use. This type of probability is usually referred to as a pseudo-random number generator (PRNG). 

The formula for finding P(X) is X = PRNG(X). The exact formula will vary depending on the PRNG you are using.

Probability examples:

Example 1: Given a bag containing eight marbles, four white and four blue, if you randomly pick two marbles simultaneously (without replacing them), what is the probability of choosing two white marbles?

Answer: 

The sample space is W = {WW, Wb, Wg, Wg} and B = {Bb, Bg, bB}. 

We have to find the number of ways that both marbles are white,

P = 1/4 × 1/4= 1/16

The event of “both marbles being white” is impossible. The event has been removed from the sample space. We have to find the probability of choosing an even number or an odd number of black marbles.

The sample space is B = {Bb, Bg, bB}.

P = 1/4 × 1/4= 1/16

All the outcomes are possible (even numbers or odd numbers). The probability of choosing either an even or odd number is:

P(even) = P(odd) or P(odd) = P(even)/2

Example 2: Given a bag containing two white marbles, four black marbles, and six green marbles, if you randomly pick two marbles simultaneously (without replacing them), what is the probability of choosing two white marbles?

Answer: 

First, we have to find the number of ways that both marbles are white. 

Then there are 1/2 × 1/2=1/4 combination ways that are white out of two marble chosen. 

P = ¼.

We have to find the probability of choosing an even number or an odd number of black marbles. 

The sample space is W = {W, w, w} and B = {Bb, Bg, bB}. 

P(even) = P(odd) or P(odd) = 1/2/2

We have to find the possible outcomes of both two marble chosen to be black.

P = 1/4 × 1/2=1/8

Conclusion: 

Calculating the probability of an event is typically straightforward. First, you’ll need to determine the type of probability and then locate the appropriate formula based on that type. This can sometimes be tricky to figure out because of the variety of probability. The probability of an event, such as rolling a six on a die. It’s the chance that you could roll a six on a die. Therefore, P(A)=1-P(not A) would be 1-(0.5). So using probability formula, with P(A)=1-(0.5)=0.5, it’s a 5050 chance of a six when you roll the die.