# Probability, Objectives, Events and Their Probability, Studying this Lesson

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In this lesson we shall discuss some basic concepts of probability, addition theorem, dependent and independent events, multiplication theorem, Bayes’ theorem, random variable, its probability distribution and binomial distribution.

## Objectives

After studying this lesson, you will be able to:

Define probability of occurrence of an event;

Cite through examples that probability of occurance of an event is a non-negative fraction, not greater than one;

Use permutation and combinations in solving problems in probability;

State and establish the addition theorems on probability and the conditions under which each holds;

Generalize the addition theorem of probability for mutually exclusive events;

Understand multiplication law for independent and dependent events and solve problems related to them.

Understand conditional probability and solve problems related to it.

Understand Baye’s theorem and solve questions related to it.

Define random variable and find its probability distribution.

Understand and find, mean and variance of random variable.

Understand binomial distribution and solve questions based on it.

## Events and Their Probability

In the previous lesson, we have learnt whether an activity is a random experiment or not. The study of probability always refers to random experiments. Hence, from now onwards, the word experiment will be used for a random experiment only.

In the proceeding lesson, we have defined different types of events such as equally likely, mutually exclusive, exhaustive, independent and dependent events and cited examples of the above mentioned events.

Here we are interested in the chance that a particular event will occur, when an experiment is performed. Let us consider some examples.

What are the chances of getting a ‘ Head’ in tossing an unbiased coin? There are only two equally likely outcomes, namely head and tail. In our day to day language, we say that the coin has chance in of showing up a head. In technical language, we say that the probability of getting a head is .

Similarly, in the experiment of rolling a die, there are six equally likely outcomes, or . The face with number (say) has chance in of appearing on the top. Thus, we say that the probability of getting is .

In the above experiment, suppose we are interested in finding the probability of getting even number on the top, when a die is rolled. Clearly, the possible numbers are and and the chance of getting an even number is in. Thus, we say that the probability of getting an even number is , i.e. .

The above discussion suggests the following definition of probability.

If an experiment with exhaustive, mutually exclusive and equally likely outcomes, outcomes are favourable to the happening of an event , the probability of happening of is given by

Since the number of cases favourable to the non-happening of the event are , the probability that ’ will not happen is given by

[Using (1)]

Obviously, as well as are non-negative and cannot exceed unity.

i.e.

Thus, the probability of occurrence of an event lies between and [including and ].

**Note:**

1. Probability of the happening of an event is known as the probability of success and the probability of the non-happening of the event as the probability of failure.

2. Probability of an impossible event is 0 and that of a sure event is 1

If , the event is certainly going to happen and

If , the event is certainly not going to happen.

3. The number of favourable outcomes to an event cannot be greater than the total number of outcomes.

Example 1:

What is the chance that a leap year, selected at random, will contain Sundays?

Solution:

A leap year consists of days consisting of weeks and extra days. These two extra days can occur in the following possible ways.

(1) Sunday and Monday

(2) Monday and Tuesday

(3) Tuesday and Wednesday

(4) Wednesday and Thursday

(5) Thursday and Friday

(6) Friday and Saturday

(7) Saturday and Sunday

Out of the above seven possibilities, two outcomes,

e.g., (1) and (7), are favourable to the event

Sundays

Example 2:

In a single throw of two dice, what is the probability that the sum is ?

Solution:

The number of possible outcomes is . We write them as given below:

Now, how do we get a total of We have:

In other words, the outcomes and are favourable to the said event,

i.e., the number of favourable outcomes is .

Hence, (a total of)