Probability (revision)

Probability (first glance):

Probability is a measure of certainty (surety) of an event. It takes a value between

‘0’ and ‘1’. For an impossible event, the probability is zero. For a sure event, the
probability is one.

Example: If a regular (fair) coin is tossed, the probability that a ‘head’ occurs is 1/2
= 0.50 (50% chance).

Example: If a fair dice is tossed, the probability that a ‘3’ occurs is 1/6 = 0.167
(16.7% chance).

It should be noted here that in old English, ‘die’ is singular and ‘dice’ is plural. In
present-day English, ‘dice’ is both singular and plural.

Deterministic and random experiments:

An experiment that has only one possible outcome is a ‘deterministic’ experiment.

Example: Counting the number of stairs of a particular building is a deterministic

experiment. If we do the experiment repeatedly, we will get the same result (if we
do not make a mistake).

A ‘random’ experiment has more than one possible outcome.

Example: Tossing a coin, throwing a dice, counting the number of calls received in
an hour, etc. are random experiments.

• Probability is associated with random experiments.

Sample space:

The set of all possible outcomes of a random experiment is called the sample space.
It is usually denoted by 𝑆𝑆. It is comparable to the universal set in set theory.

Example: In coin tossing experiment, 𝑆𝑆 = {𝐻𝐻, 𝑇𝑇}.

Example: In dice throwing experiment, 𝑆𝑆 = {1, 2, 3, 4, 5, 6}.

Example: In a cricket match, 𝑆𝑆 = {win, loss, tie, postponed, cancelled}.

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Outcomes (elements) in S should be mutually exclusive (two outcomes cannot occur
together). Also, S should be exhaustive (complete, i.e., no possible outcome should
be left out).

Event:

Any subset of the sample space is called an event. When a random experiment is
going to be conducted, we are interested in the probabilities of different events.

Example: In dice throwing, 𝐴𝐴 = {1}, 𝐵𝐵 = {1, 4, 5}, etc. are events. Note that B
denotes the event that 1 or 4 or 5 occurs. They cannot happen together!

Example: In cricket match, 𝐶𝐶 = {loss}, 𝐷𝐷 = {win, tie}, etc. are events.

Events are sets. So, upper-case letters should be used for notation.

Operations on events:

𝐴𝐴 ∪ 𝐵𝐵 occurs when 𝐴𝐴 or 𝐵𝐵 (or both) occur.

𝐴𝐴 ∩ 𝐵𝐵 occurs when both 𝐴𝐴 and 𝐵𝐵 occur.
𝐴𝐴𝑐𝑐 occurs when 𝐴𝐴 does not occur.

Classical or mathematical definition of probability:

Let a random experiment have 𝑛𝑛 possible outcomes that are mutually exclusive,
exhaustive and equally likely (all the outcomes have same chance). If 𝑚𝑚 of these
outcomes are favorable to an event 𝐴𝐴, then probability of 𝐴𝐴 is given by:
𝑚𝑚
𝑃𝑃(𝐴𝐴) =
𝑛𝑛
Note that, we cannot use this formula when all the outcomes are not equally likely,
or the total number of possible outcomes, 𝑛𝑛, is infinite.

Example: Let a fair dice be thrown. Let 𝐴𝐴 = {1, 4, 5}. Then

𝑚𝑚 3
𝑃𝑃(𝐴𝐴) = = = 0.5.
𝑛𝑛 6
Example: Consider a cricket match. 𝑆𝑆 = {win, loss, tie, postponed, cancelled}.
2
Let 𝐷𝐷 = {win, tie}. Then, we should NOT say 𝑃𝑃(𝐷𝐷) = . (Why not?)
5

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Empirical or statistical or frequency definition of probability

Empirical means observation-based, i.e., data-based.

Let an experiment be conducted 𝑛𝑛 times, where 𝑛𝑛 is large. If an event 𝐴𝐴 occurs 𝑓𝑓𝐴𝐴

times, then
𝑓𝑓𝐴𝐴
𝑃𝑃(𝐴𝐴) ≈
𝑛𝑛
Here, probability is approximately equal to relative frequency. We cannot use this
formula when the experiment cannot be repeated under the same conditions. Also,
𝑓𝑓𝐴𝐴
𝑃𝑃(𝐴𝐴) = lim
𝑛𝑛→∞ 𝑛𝑛

Example: An unfair dice is tossed 1000 times. “6” occurred 400 times, Then,
400
𝑃𝑃(6) ≈ = 0.40
1000
• We have not followed set notation to write the event ‘6’. Some good books
have also done that for comfort.

Subjective definition of probability

Sometimes probability is someone’s judgement or belief.

Example: Consider the statement: “There is a 90% chance (probability 0.90) that I’ll
get an ‘A’ in this course.” Here, the probability 0.90 shows the judgement of the
‘subject’ (the person who made the statement).

We cannot use mathematical definition in the above example, because ‘getting A’

and ‘not getting A’ are not equally likely. Also, we cannot use statistical definition
here, because we cannot repeat the experiment (taking the course) under the same
conditions.

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Axioms of probability

Probability follows the following three axioms:

1. 0 ≤ 𝑃𝑃(𝐴𝐴) ≤ 1.

2. 𝑃𝑃(𝑆𝑆) = 1.

3. When 𝐴𝐴 and 𝐵𝐵 are mutually exclusive events, i.e., 𝐴𝐴 ∩ 𝐵𝐵 = ∅, then

𝑃𝑃(𝐴𝐴 ∪ 𝐵𝐵) = 𝑃𝑃(𝐴𝐴) + 𝑃𝑃(𝐵𝐵).

Explanation of Axiom 3:
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Let a fair die be thrown. Let 𝐴𝐴 = {1, 2, 3} and 𝐵𝐵 = {4, 5}. Then, 𝑃𝑃(𝐴𝐴) = and
6
2 5
𝑃𝑃(𝐵𝐵) = . Also, 𝐴𝐴 ∪ 𝐵𝐵 = {1, 2, 3, 4, 5}, so that 𝑃𝑃(𝐴𝐴 ∪ 𝐵𝐵) = = 𝑃𝑃(𝐴𝐴) + 𝑃𝑃(𝐵𝐵). If 𝐴𝐴
6 6
and 𝐵𝐵 are not mutually exclusive, equality will not hold.

Some important theorems:

1. 𝑃𝑃(∅) = 0.

2. 𝑃𝑃(𝐴𝐴𝑐𝑐 ) = 1 − 𝑃𝑃(𝐴𝐴).

3. Addition rule of probability: For any two sets 𝐴𝐴 and 𝐵𝐵,

𝑃𝑃(𝐴𝐴 ∪ 𝐵𝐵) = 𝑃𝑃(𝐴𝐴) + 𝑃𝑃(𝐵𝐵) − 𝑃𝑃(𝐴𝐴 ∩ 𝐵𝐵).

Exercise:

In a community, 25% of the families have cars, 15% have washing machines and
10% have both. A family is selected at random from the community. What is the
probability that the family has (i) a car or a washing machine? (ii) neither a car nor
a washing machine?

Solution: Do it yourself.