Probability in Maths

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Probability is a branch of mathematics that deals with calculating the likelihood of a given event’s occurrence. It is expressed as a number between 0 and 1.

Let’s learn about the basic probability concepts, their formulas, types and examples.

Probability Definition

Probability is the measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1.

The values of probability range from 0 to 1, with 0 telling us improbability and 1 denoting certainty.

For calculating probability, we simply divide the number of favorable outcomes by the total number of outcomes.

Probability Theory

Probability theory is a branch of mathematics that deals with the interpretation of random events and the likelihood of these events occurring.

Probability theory starts with basic concepts such as random experiments, sample spaces, events, and the probability of events.

A random experiment is any process or action that results in one of several possible outcomes, like tossing a coin or rolling a die. The sample space is the set of all possible outcomes of an experiment, and an event is a specific outcome or a set of outcomes.

Read More On Probability Theory

Let’s discuss the basic concepts of Probability Theory in some detail.

Basic Probability Concepts

Here are some basic concepts or terminologies used in probability:

1. Sample Space

The set of every possible outcome in a probability experiment(trail)is known as the sample space. For example, the sample space for tossing a coin is head and tail.

2. Sample Point

It is one of the possible results. Consider the example of rolling a fair six-sided dice. The sample space for this experiment consists of six possible outcomes: 1,2,3,4,5 or 6. Each of these possible outcomes is a sample point.

3. Experiment

A process or trial that provides a range of potential results is called an experiment. Any scenario where the results are uncertain may be the subject of the experiment. For example, The tossing of a coin, Selecting a card from a deck of cards, throwing a die, etc.

4. Event

A subset of the sample space. For example: getting 1 as an outcome while rolling a die is an event.

5. Favorable outcome

An occurrence that has produced the desired consequence or an expected event.

Probability of an Event

The probability of an event is a measure of the likelihood that the event will occur, expressed as a number between 0 and 1.

An event with a probability of 1 is considered certain to happen, while an event with a probability of 0 is certain not to happen.

Let’s discuss different types of events in probability.

Equally Likely Events

Equally likely events are those whose chances or probabilities of happening are equal. Both events are not related to one another. For example, there are equal possibilities of receiving either a head or a tail when we flip a coin.

Exhaustive Events

We call an event exhaustive when the set of all experiment results is the same as the sample space.

Mutually Exclusive Events

Events that are mutually exclusive cannot occur at the same time. For instance, the weather may be hot or chilly simultaneously. We can’t have the same weather at the same time.

Complimentary events

The non-occurring events. The complement of an event A is the event, not A(or A’).

Sample Space Illustration in Probability

Probability Formula

P(E) = Number of favorable outcomes / Total number of outcomes

Note :where P(E) denotes the probability of an event E.

This formula defines the possibility of an event occurring. It is defined as the ratio of the number of favorable outcomes to the total number of outcomes.

Probability Formula Examples

Example 1. There are 8 balls in a container, 4 are red, 1 is yellow and 3 are blue. What is the probability of picking a yellow ball?

Probability of picking a ball

Answer:

The probability is equal to the number of yellow balls in the container divided by the total number of balls in the container, i.e. 1/8.

Example 2: A dice is rolled. What is the probability that an even number has been obtained?

Probability of Rolling a Dice

Answer:

When fair six-sided dice are rolled, there are six possible outcomes: 1, 2, 3, 4, 5, or 6.

Out of these, half are even (2, 4, 6) and half are odd (1, 3, 5). Therefore, the probability of getting an even number is:

P(even) = number of even outcomes / total number of outcomes

P(even) = 3 / 6

P(even) = 1/2

Probability Tree Diagram

A tree diagram in probability is a graphic representation that helps us in determining the likely outcomes that is whether an event will occur or not. It helps us understand the all possibilities of an event and which possibilities can occur and cannot occur.

Probability Tree Diagram

Types of Probability

These are 3 major types of probability :

Theoretical Probability
Experimental Probability
Axiomatic Probability

Theoretical Probability

It is focused on the likelihood of anything occurring. The reasoning behind probability is the foundation of scientific probability.

If a coin is flipped, the statistical chance of having a head is 1/2. An event’s statistical chance is the chance that the occurrence will occur. It is determined by dividing the number of desirable outcomes by the total number of outcomes is thus termed theoretical probability.

Learn more about Theoretical Probability with the help of the following articles:

Experimental Probability

It is founded on the results of an experiment. The experimental chance can be determined by dividing the total number of trials by the number of potential outcomes.

For example, if a coin is flipped ten times and heads are reported six times, the experimental chance of heads is 6/10 or 3/5. Let’s discuss the experimental probability in the following articles:

Axiomatic Probability

A collection of laws or axioms that apply to all forms is established in axiomatic probability. Kolmogorov developed these axioms, which are known as Kolmogorov’s three axioms.

The axiomatic approach to probability quantifies the likelihood of events occurring or not occurring.

Following are the articles based on the axiomatic probability:

Probability Theorems

The theorems of probability are helpful to understand the applications of probability and also perform the numerous calculations involving probability.

Some of the most important theorems of probability are :

Theorem 1: The sum of the probability of happening an event and not happening an event is equal to 1. P(A) + P(A’) = 1

Theorem 2: The probability of an impossible event or the probability of an event not happening is always equal to 0.

P(ϕ) = 0

Theorem 3: The probability of a sure event is always equal to 1.

P(A) = 1

Theorem 4: The probability of happening of any event always lies between 0 and 1.

0 < P(A) < 1

Theorem 5: If there are two events A and B, we can apply the formula of the union of two sets and we can derive the formula for the probability of happening of event A or event B as follows.

P(A∪B) = P(A) + P(B) − P(A∩B)

Also for two mutually exclusive events A and B, we have P( A U B) = P(A) + P(B)

We have a few more formulas used to calculate the probability of any event :

1. Whenever an event is the union of two events, say A and B, then

P(A or B) = P(A) + P(B) – P(A∩B)

P(A ∪ B) = P(A) + P(B) – P(A∩B)

2. Whenever an event is the complement of another event if A is an event, then

P(not A) = 1 – P(A) or P(A’) = 1 – P(A).

P(A) + P(A′) = 1.

3. When event A has already taken place and the probability of event B is to be calculated, then

P(B, given A) = P(A and B), P(A, given B)

Similarly, it can be vice versa in the case of event B.

P(B∣A) = P(A∩B) / P(A)

4. Whenever an event is the intersection of two events, that is, events A and B need to occur simultaneously. Then

P(A and B) = P(A)⋅P(B)

P(A∩B) = P(A)⋅P(B∣A)

Bayes’ Theorem on Conditional Probability

Bayes’ Theorem defines the probability of an event based on the condition of occurrence of other events. It is also called conditional probability.

It helps in calculating the probability of occurrence of one event based on the condition that another event is also taking place or occurring.

It is named after Thomas Bayes who was an English statistician and is used widely in machine learning, mathematics, and statistics.

For example, let us assume you have two bags of marbles in various colors. Bag A has three red and two blue marbles, while bag B has one red and four blue marbles. You pick one of the bags at random, then you pick a marble at random from that bag. Given that you selected a red marble, what is the probability that you selected bag A? Such probability is called conditional probability.

Bayes’ Theorem Formula

Following is the Bayes’ Theorem Formula :

P(A|B) = P(B|A)⋅P(A) P(B)

where:

P(A|B) denotes how often event A happens on a condition that B happens.
P(B|A) denotes how often event B happens on a condition that A happens.
P(A) is the probability of event A occurring.
P(B) is the probability of event B occur

Common Probability Problems

We will now see some common probability problems that are given in school tests. This will prepare you for the questions and you can understand the methodology to solve them.

1. Probability of Tossing Coin

Now let us take into account the case of coin tossing to understand probability in a better way.

A) Tossing a Coin

A single coin when flipped has two possible outcomes, a head or a tail. The definition of probability when applied here to find the probability of getting a head or getting a tail.

The total number of possible outcomes = 2

Sample Space = {H, T} H: Head, T: Tail

P(H) = Number of Heads/ Total Number of outcomes = 1/2

P(T) = Number of Tails/ Total Number of outcomes = 1/2

B) Tossing Two Coins

There are a total of four possible results when tossing two coins. We can calculate the probability of two heads or two tails using the formula.

The probability of getting two tails can be calculated as :

Total number of outcomes = 4

Sample Space = {(H,H),(H,T),(T,H),(T,T)}

P(2T) = P(0H) = Number of outcomes with two tails/ Total number of outcomes = 1/4

P(1H) = P(1T) = Number of outcomes with one head/ Total number of outcomes = 2/4 = 1/2

C ) Probability of Tossing Three Coins

The number of total outcomes on tossing three coins is 2³= 8. For these outcomes, we can find the probability of various cases such as getting one tail, two tails, three tails, and no tails, and similarly can be calculated for several heads.

Total number of outcomes = 2³= 8

Sample space = {(H, H, H), (H, H, T),(H, T, H), (T, H, H), (T, T, H), (T, H, T), (H, T, T), (T, T, T)}.

P(3T) = P(0 H) = Number of outcomes with three tails/ Total Number of outcomes = 1/8

2. Probability of Rolling Dice

Various games use dice to decide the movements of the player during the games. A dice has six outcomes. Some games are played using two dice. Now let us calculate the outcomes, and their probabilities for one dice and two dice respectively.

Rolling One Dice

The number of outcomes is 6 when a die is rolled and the sample space is = {1,2,3,4,5,6}.

Let us now discuss some cases.

P(Even Number) = Number of outcomes in which even number occur/Total Outcomes = 3/6 = 1/2

P(Prime Number) = Number of prime number outcomes/ Total Outcomes = 3/6 = 1/2

Rolling Two Dice

The number of total outcomes, when two dice are rolled, is 6²= 36.

Probability of Rolling Two Dice

Let us check a few cases in the above example,

Probability of getting a doublet(Same number) = 6/36 = 1/6

Probability of getting a number 3 on at least one dice = 11/36

3. Probability of Cards

Spades, clubs, diamonds, and hearts make up the four suits that form a deck of 52 playing cards. There are a total of 52 cards, with 13 in each of the four suits (clubs, diamonds, hearts, and spades). The symbols for the cards are listed below.

Probability of Drawing Cards

Black cards are spades and clubs. Red cards are hearts and diamonds. Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, queen, and King are the 13 cards that make up each suit. The jack, queen, and king are referred to as face cards in these.

Probability of getting a black card is P(Black Card) = 26/52 = 1/2

Probability of getting a heart is P(Heart) = 13/52 = 1/4

Probability of getting a face card is P(Face card) = 12/52 = 3/13

Probability Density Function

Probability density function is defined as a function that contains all the possible outcomes of any given situation.

A Probability Density Function is a fundamental concept in statistics and probability theory. It used to specify the probability of a continuous random variable falling within a particular range of values.

Unlike discrete random variables which have probabilities associated with specific points, continuous random variables require a function to define their probabilities over an interval.

It is also known as the Probability Distribution Function.

Probability Applications

Some of the real life applications of probability are:

1. Probability in Weather Forecasting :

Probability is used to forecast weather. Meteorologists use probability to find patterns in weather conditions and predict weather. They use many advance scientific tools to collect and analyze data. By using probability they can predict rain, storms, or snow.

2. Probability in Finance :

Traders and Financial experts use probability models to determine the outcomes of stock market. Using probability they can assess market risksand make better investment decisions. It is also used to predict future values of stocks.

3. Probability in Healthcare :

Probability is widely used in healthcare sector to assess the risk of a disease and make plans for patient care. For example, in recent 2019 covid pandemic, various drugs were tested and scientist could tell the chances of it working on a patient.

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Probability Examples and Solutions

We have provided you with some probability problems with their solutions.

Problem 1. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5?

Solution:

Here S is {1,2,3,4,5,6,7,8………19,20}.

Favorable outcomes(E) = {3,5,6,9,10,12,15,18,20}

P(getting multiple of 3 or 5) = n(E)/n(S) = 9/20.

Problem 2. In a lottery, there are 10 prizes and 25 blanks. A lottery is drawn at random. What is the probability that a blank is selected?

Solution:

S= Total outcomes

E= getting a blank.

n(E) = 25

P(Getting a blank) = n(E)/n(S) = 25/35 = 5/7.

Problem 3. A bag contains 4 white, 5 red, and 6 blue balls. Three balls are drawn at random from the bag. The probability that all of them are red, is:

Solution:

Let S be the sample space.

Then, n(S) = Number of ways of drawing 3 balls out of 15

= ¹⁵C₃= 455

Let E = event of getting all the 3 red balls.

n(E) = ⁵C₃= 10

P(E) = n(E)/n(S) = 10/455 = 2/91.

Problem 4. In a class there are 10 girls and 15 boys, what is the probability that 1 girl and 2 boys are selected?

Solution:

Let S be the sample space.

Then, n(S)= Number of ways of selecting 3 children out of 25

= ²⁵C₃

= 2300.

Let E= event of selecting 1 girl and 2 boys.

n(E) = ¹⁰C₁*¹⁵C₂= 1050

P(E) = n(E)/n(S) = 1050/2300 = 21/46.

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