According to mathematics, probability is defined as the possibility of any event happening. Several events can’t be predicted with utmost certainty, but the formulas and theories of probability help us reach the outcome. It helps in predicting the chances of any event from happening. In this chapter, students are taught the simple problems on single events initially, but as the topics progress, you get to learn more challenging things. 

The probability of any event cannot be less than 0 or more than 1. It means the possibility of any event from happening will never be less than 0 and cannot be more than 1. It can also be said that all the sample spaces of an event will always add up to 1.

It is easy to find the probability of any event when simple problems on single events are given. But on the other hand, here is what you can do when multiple events are given. 

Probability of event to happen P(E) = Number of favourable outcomes/Total Number of outcomes

It is the basic formula that helps determine an event’s probability, but other formulas are used in different situations. People usually confuse between desirable outcomes and favourable outcomes.

Probability is categorised into three major types:

• Theoretical probability

The theoretical probability can be defined as possible outcomes of any event to happen. This type of probability depends mainly on the reasoning behind the concept of probability. For example, when you toss a coin, the probability of getting a tail or head is fixed, which is ½ or 50%. 

• Experimental probability

As the name suggests, this is the category where the probability of any event to happen is based on an experiment’s observations. The experimental probability can be determined by dividing the number of possible outcomes by the total number of outcomes. 

• Axiomatic probability  

The axiomatic probability is the one that is governed by a specific set of rules or so-called axioms which apply to all types. These axioms are popularly known as Kolmogorov’s three axioms.

Let’s assume that an event ‘E’ can occur in ‘r’ ways and the total number of outcomes be ‘n’. The equation of probability can be expressed as:

P(E)= r/n

The probability of the event not occurring is expressed as:

P(E’)= (n-r) /n = 1- (r/n), where E’ is the probability of event not happening

We can also say, 

P(E)+ P(E’)= 1

• Sample space
• It is defined as the set of possible outcomes that occur in an event.
• Sample point
• It is defined as one of the possible results of an event.
• Trial or experiment
• It is an event or trial where the outcomes are uncertain and only predicted.
• Event
• It is defined as the single outcome of any experiment.
• Outcome
• It is defined as the possible or probable result of any event or experiment.
• Complimentary event
• It is defined as the non-happening events in the experiment.

Example 1: What will be the probability of getting four on rolling a die.

Ans:

Sample Space = {1, 2, 3, 4, 5, 6}

Number of favourable event = 1

i.e. {4}

Total number of outcomes = 6

Thus, Probability, P = ⅙.

Example 2: A bag contains five blue balls, four red balls and 11 white balls. If you draw out three balls randomly from the bag, what is the probability that the first ball will be red, the second ball would be blue, and the third ball would be white?

Ans:

The probability of getting the first ball being red is 4/20.

Now, 

We have drawn a ball for the first event to occur, which means the number of possibilities left for the second event will be 20 – 1 = 19.

Therefore, the probability of getting the second ball being blue is 4/19.

With two events already done, the number of possibilities left for the third event is 19 – 1 = 18.

The probability of the third ball being white is 11/18.

Therefore, the probability is 5/20 x 4/19 x 11/18 = 44/1368 = 0.032.

Example 3: A coin is tossed thrice. What is the probability of getting at least one head?

Ans: 

Sample space or total number of possible outcomes= [HHH, HHT, HTH, THH, TTH, THT, HTT, TTT]

Total number of ways = 2 × 2 × 2 = 8. Fav. Cases = 7

P (A) = ⅞.