Probability

Probability is an experimental approach to the game of chances we come across every day in our lives. It is a way to describe these “chances” mathematically and then analyze them. It allows us to measure the chances of something occurring. 

The concept of probability was developed by famous philosopher and mathematician Blaise Pascal in the 17th century.

Read in Detail: Probability in Maths | Formula, Theorems, Definition, Types

Let’s do an experiment, 

Suppose we have a coin, we toss it a number of times and then collect the data about heads and tails. We will perform this experiment multiple times and record the data in a table. 

Let’s calculate the values of two fractions given below, 

Now if we keep on calculating these values, we will notice that the values start converging towards 0.5 as the number of experiments trials increases. 

Trial

 A trial is an action that results in one or more outcomes

In the given experiment, tossing a coin is a trial. 

Event

 An event is an outcome of the trial

Getting a head or a tail after tossing the coin can be considered an event in our experiment. Now let’s define probability. 

Sample Space

It is the set of all the possible outcomes

Since sample space, consists of all the possible outcomes. In our case, the number of heads can vary from 0 to the number of times the coin is tossed.

{0, 1, 2, 3, …..n}

Probability

It is a value that denotes the chances of occurrence of some event. 

Let “n” be the total number of trials, and E be an event. The probability of occurrence of that event is, 

Notice that by its definition, the numerator will always be less than or equal to the denominator. So, 

P(E) ≤ 1

Question: Based on the experiment above, let us say a coin was tossed 20 times, and we obtained 15 heads and 5 tails. Find the probability of getting a head and a tail when this coin is tossed again. 

Solution: 

We know that the coin was tossed 20 times, suppose we want to calculate the probability for heads first. 

Probability for Heads: 

So the event that we will look for is getting head, this happened 15 times when the coin was tossed 20 times. 

Probability for Tails: 

So the event that we will look for is getting a head, this happened 5 times when the coin was tossed 20 times. 

Notice that in the previous example, the probability of head and probability of tails when summed up, give us 1. 

Addition Rule for Probability: Probability theory is a cornerstone of modern mathematics, used extensively in fields ranging from statistics and economics to physics and engineering. At its core, probability theory deals with uncertainty, allowing us to quantify the likelihood of different outcomes in various scenarios. One of the fundamental principles in probability theory is the Addition Rule. This concept provides a systematic way to calculate the probability of the union of two or more events.

The General Addition Rule for Probability is given by P(A or B) = P(A) + P(B) – P(A and B) where A and B are the two events. For mutually exclusive events, P(A and B) = 0. So P(A or B) = P(A) + P(B) for mutually exclusive events.