# Important Probability formulas and rules of Probability

## Introduction

Probability is a branch of statistical mathematics. It deals with the likeliness of an event to occur in numerical descriptions. The probability of an event lies between 0 and 1. If the probability indicates 0, that means the chances of the event is not possible. While 1 denotes the probability of the event to occur is maximum. There are different probability formulas and rules, which we will discuss in the following paragraphs.

Probability in non-mathematical terms resembles ‘Chances’. If the probability is on the higher side, then the chances of the event are higher. Likewise, if the probability of an event shifts to the lower side, then the chances of that event occurring are less.

For instance, a coin is tossed, the outcome of the toss is only two. It can be either a head or a tail. So, the probability of getting ahead is 0.5. Similarly, the probability of getting a tail is also 0.5.  However, Probability as a whole subject is not that simple. It has complicated formulas, equations, numerical terms and so on.

Probability theory is applied in day to day life. It is applied in simple daily predictions as well as in complicated industrial calculations. It is used in risk assessment and modelling. Also, the insurance industry and markets use actuarial science to determine prices and make trading decisions. On the other hand, government apply probability in environmental regulation, entitlement analysis and financial regulation.

You can also read our article on Scope of Statistics.

## Some common terms

• Probability of an event is denoted by ‘P’.

P = number of favourable outcomes/ total number of outcomes

For example: When is coin is tossed, what is the probability of getting ahead?

Solution: Number of favorable outcomes (getting a head) = 1

Total number of outcomes = 2

We know that, the probability formulas say

P (E) = n (E) / n(S)

Probability of getting a head = ½ = 0.5 Or 50%

• Random experiment

It is not possible to predict an outcome in advance. Also, It has more than one outcome.

• Outcome

It is a possible result of an experiment or trial. In addition, each possible outcome is unique.

• Sample space (S)

A set of possible outcomes is called sample space.

• An event

It is any combination of outcomes.

• Equally likely

It means that each outcome of an experiment occurs with equal probability.

• Probability function

It helps to obtain the probability of every outcome.

January 27, 2022

## Types of Probability

It is broadly classified into three parts.

• Classical Probability

It is a simple form of probability. It is an approach to identify and assume that any random process has a set of outcomes. And the possibility of each outcome to occur is equally likely.

Then, the probability formulas of an event say

P (E) = n (E) / n(S)

Here, P (A) indicates the probability of event A. While n (E) indicates a number of possible outcomes. And n (S) means the total number of possible outcomes.

• Relative Frequency Definition

It is an estimate for probability events. the formula for relative frequency is given below

Relative Frequency = frequency of an event/total number of frequencies

• Subjective Probability

It is a type of probability that comes out of an individual’s self-judgment or own experience about a specific event. It has no calculation or formulas to follow.

## What are the 5 rules of probability?

Let’s move and learn something more about probability, its rules and probability questions and examples.

##### Rule 1

The first rule of probability reflects the very basics of the topic. It informs us that the likelihood of an event lies between 0 and 1. In that case, 0 means that the chances of the event occurring is not possible. While 1 denotes that the chances of the event to occur is maximum.

In other words, the probability of an outcome A; denoted as P (A), is a number between 0 and 1, indicating the proportion of the time that the outcome A occurs in the long span of time. For instance, when the probability of an event is close to 0, its occurrence is unlikely. If the probability of an event is 0.5, there is about a 50% chance that the event will occur, and when the probability of an event is close to 1, the event is highly likely to occur.

This is denoted by 0<= P (A) <=1

##### Rule 2

The sum of the probabilities of all the possible outcomes is 1. For instance, in rolling a single dice, each outcome in the sample space has a probability of ⅙, and the total of all outcomes is equal to 1.

##### Rule 3

If an event A cannot occur, i.e., the event contains no members in the sample space, its probability is 0. For instance, when a single dice is rolling, the probability of getting a number greater than 6. Since all the sample spaces are 1,2,3,4,5, and 6. As it is seen that all the space is less than 6. So, the probability of getting a number more than 6 is zero i.e., not possible.

##### Rule 4

If an event A is certain, its probability is 1. For instance, when a single dice is rolling, then what shall be the probability of getting a number less than 8. Since all the outcomes are 1,2,3,4,5 and 6. As it is seen that all the outcomes are less than 8. So, the probability is 1.

P (number less than 8) = 6/6 = 1 or 100%

Then event of the number getting less than 8 is certain.

##### Rule 5

The Complement Rule states that the sum of the probabilities of an event and its complement must equal 1.

P(A)+P(A′)=1

So, P (A’) = 1 – P (A)

## Important probability formulas

The probability formulas help to find the ratio of the number of favourable outcomes to the total number of possible outcomes.

The probability formulas of an event = number of favourable outcomes/total number of possible outcomes.

P (A) = n (E)/n (S)

P (A) < 1

Here, P (A) indicates the probability of event A. While n (E) indicates the number of possible outcomes. And n (S) means the total number of possible outcomes.

Besides, if the probability of occurring an event is P (A). Then, the probability of not occurring the event P (A’) is as follows

P (A’) = 1 – P (A)

##### Example 01:.

Probability of obtaining an odd number when a dice is rolling for one time.

##### Solution:

Sample space (S) = {1,2,3,4,5,6}

Total number of possible outcomes n (S) = 6

Favorable outcomes = {1,3,5}

So, n (E) = 3

We know that, the probability formulas say

P (A) = n (E)/n (S)

P (getting an odd number) = 3/6 = ½ = 0.5

##### Example 02:

Probability of getting head at least once on tossing a coin twice.

##### Solution:

Sample Space (S) = {HH, HT, TH, TT}

Here, H denotes Head and T denotes Tail.

Favorable outcomes = {HH, HT, TH}

Therefore, n (S) = 4

and n (E) = 3

we know that the probability formulas say

P (A) = n (E)/n (S)

P (getting Head at least once on tossing a coin twice) = 3 / 4 = 0.75

This means that the chances of getting at least one Head on tossing a coin twice are 0.75

## Important list of Probability Formulas

Event (A OR B)

Also given by P (A U B) = P (A) + P (B) – P (A ∩ B)

If A & B are two mutually exclusive events then P (A ∩ B) = 0 and P (A U B) = P (A) + P (B).

For example,

A = {Numbers greater than or equal to 4 in a dice roll} = {4, 5, 6}

B = {Numbers lesser than or equal to 4 in a dice roll} = {1, 2, 3, 4}

Thus, (A U B) = P (A) + P (B) = {1, 2, 3, 4, 5, 6}

• Event (A AND B)

Also given by P (A ∩ B) = P (A).P (B)

It gives the common elements that form the individual subsets of events A and B. For example,

A = {Numbers greater than or equal to 4 in a dice roll} = {4, 5, 6}

B = {Numbers lesser than or equal to 4 in a dice roll} = {1, 2, 3, 4}

Thus, (A ∩ B) = P (A) . P (B) = {4}

• Event (A but NOT B)

For example,

A = {Numbers greater than or equal to 4 in a dice roll} = {4, 5, 6}

B = {Numbers lesser than or equal to 4 in a dice roll} = {1, 2, 3, 4}

Thus, (A but NOT B) = A – B = {1, 2, 3}; elements common in A and B get eliminated from A.

• Event (B but NOT A)

For example,

A = {Numbers greater than or equal to 4 in a dice roll} = {4, 5, 6}

B = {Numbers lesser than or equal to 4 in a dice roll} = {1, 2, 3, 4}

Thus, (B but NOT A) = B – A = {5, 6}; elements common in A and B get eliminated from B.

• Event (NOT A)

If probability of occurrence of an event, P (A) then, the probability of non-occurrence of the same event is P(A’). Some probability formulas based on them are as follows:

P(A.A’) = 0

P(A.B) + P (A’.B’) = 1

P(A’B) = P(B) – P(A.B)

P(A.B’) = P(A) – P(A.B)

P(A+B) = P(AB’) + P(A’B) + P(A.B)

• Conditional Probability

P(B/A): Probability of event B when event A has occurred.

P(A/B): Probability of event A when event B has occurred.

P(A ∩ B) = P (A) . P (B/A)

These formulas in the above section will help you solve probability related mathematical problems.

#### What do U and ∩ mean in probability?

‘U’ means uniform distribution in probability. On the other hand, ‘∩’ refers to the intersection of sets: Besides, the intersection of two given sets is the largest set covering all the elements common to both sets.

## Some probability questions based on probability formulas

##### Question 01:

Find the probability of getting an even number greater than or equal to 4 in a dice roll.

##### Solution:

Sample space (S) = {1, 2, 3, 4, 5, 6} and E = {4, 6}

We know that, the probability formulas say

P (E) = n (E) / n (S)

= 2 / 6

P (E) = 1 / 3

##### Question 02:

Find the probability of getting at least one ‘Head’ in a double coin toss.

##### Solution:

S = {HH, HT, TH, TT}

E = {HH, TH, HT}

We know that, the probability formulas say

P (E) = n (E) / n(S)

So, P (E) = 3 / 4

##### Question 03:

Find the probability of getting ‘3’ on rolling a die.

##### Solution:

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

Number of favorable event = 1 i.e. {3}

Total number of outcomes = 6

We know that, the probability formulas say

P (E) = n (E) / n(S)

Thus, Probability, P = ⅙

##### Question 04:

Draw a random card from a pack of cards. What is the probability that the card drawn is a face card?

##### Solution:

A standard deck has 52 cards.

Total number of outcomes = 52

Number of favourable events = 4 x 3 = 12 (considered Jack, Queen and King only)

We know that, the probability formulas say

P (E) = n (E) / n(S)

Probability, P = Number of Favorable Outcomes/Total Number of Outcomes = 12/52= 3/13.

You can also read our article on B.sc Statistics or B.sc Mathematics.

## Final Thoughts

Many businesses apply the understanding of uncertainty and probability in their business decisions. Probability formulas and theories highly assist companies in optimizing their policies and making safe decisions. One practical yet frequent use for probability related concepts is in the analysis in business and to predict future sales levels.

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