Overview: Probability

 Table of contents Definition of Probability Probability line Random Experiment Mutually Exclusive Events Equally Likely Events Independent Events Compound Events Probability Formulas

## Definition of Probability

• If an event ‘E’ can happen in ‘m’ ways & fail in ‘k’ ways out of a total of ‘n’ ways, & each of them is equally likely, then the probability of the happening of ‘E’ is .
• P(E) = m/(m+k) = m/n,     where n = (m+k)
• In other words, if a random, experiment is conducted ‘n’ times & ‘m’ of them are favorable to event ‘E’, then
• P(E) = m/n
Probability of an Event = Number of favourable outcome / Total Number of Possible outcome

## Probability line

Experiment: In probability theory, an experiment or trial (see below) is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes
Outcome: In probability theory, an outcome is a possible result of an experiment. Each possible outcome of a particular experiment is unique
Sample Space: The well-defined set of possible outcomes is known as the sample space.

## Random Experiment

A random experiment is an experiment or a process for which the outcome cannot be predicted with certainty.
Example: - Drawing 2 cards from a well shuffled pack is a random experiment while getting an Ace & a King are events.

## Mutually Exclusive Events

In probability theory, two events) are mutually exclusive if they cannot both be true or occur at the same time.
Example: The events of getting a head or a tail when a coin is tossed are mutually exclusive.

## Equally Likely Events

Equally likely events are events that have the same theoretical probability (or likelihood) of occurring.
Example:  When a die is thrown, any number from 1 to 6 may turn up. In this trial, the six events are equally likely.

## Independent Events

Two events E1 and E2 are said to be independent, if the occurrence of the event E 2 is not affected by the occurrence or the non-occurrence of the event E1.
To find the probability of two independent events that occur in sequence, find the probability of each event occurring separately, and then multiply the probabilities. This multiplication rule is defined symbolically below. Note that multiplication is represented by AND.

Multiplication Rule 1: When two events, A and B, are independent, the probability of both occurring is:
P(A and B) = P(A) · P(B)

## Compound Events

When two or more events are in relation with each other, they are known as compound events.
Example: - When a die is thrown and a coin is flipped the occurring events are called compound events.

Example 1: Find the probability of getting a head in a throw of a coin.
Sol:
When a coin is tossed we either get head or tail upwards.
So, total number of cases= 2 = n,
number of favorable cases to get H = 1 = m
P (H) = No. of favorable cases/ Total no. of outcomes = (m/n)
=1/2

Example 2: An unbiased die is rolled. Find the probability of a) Getting a multiple of 3 b) getting a prime number
Sol:

When a die is rolled we can get any one of the numbers from 1 to 6.
Total number of cases = n = 6
(a) Let event A= getting a multiple of 3
Then A= {3.6}.
Therefore m=2
P (A) = m/n = 2/6 = 1/3
(b) Let event B = getting a prime number
Then B= {2, 3, 5}.
so, m = 3
P (A) =m/n=3/6=1/2

Example 3: A card is drawn from a well-shuffled pack of 52 cards. Find the probability that
(a) Card drawn is red
(b) Card drawn is Queen
(c) Card drawn is black & king
(d) Card drawn is red & number card
(e) Card drawn is either king or queen
Sol:

Before solving this problem, let us recall the game of cards. One deck of cards contains totally 52 cards. Among them we have 13 spades & 13 clubs which are black in color, 13 diamonds & 13 hearts all of which are red. In the 13 spades, 9 are numbered cards, numbered from 2 to 10, one ace card & 3 face cards namely J, Q & K. Similarly for the clubs, diamonds & hearts also.
Totally there are 52 cards & any one can be drawn
So, total number of cases = n = 52
(a) There are 13 diamonds & 13 hearts which are red
Number of red cards = m = 26
P (getting red) =26/52=1/2
(b) There are 4 queens = 4
P (getting queen) = m/n=4/52
=1/13
(c) The king of spade & clubs are black
No. of cards which is king & black=m=2
P (king & black) =m/n =2/56
=1/26
(d) The 9 number cards of hearts & 9 number cards of diamonds are red.
No. of cards which are red & number cards = m = 18
P (red & number)=m/n=18/52
=9/26
(e) There are 4 queens & kings
No. of favorable case = m = 8
P (queen or king) = m/n = 8/52
= 2/13

Example 4: A bag contains 6 white beads & 4 red beads. A bead is drawn at random. What is the probability that the bead drawn is white?
Sol:
Total no. of beads in the bag = 6+4 =10. n= 10
Any one of the 6 white beads can be selected, m=6
Therefore, P (getting white bead) = m/n=6/10
=3/5

Example 5: A box contains 8 red marbles, 6 green marbles & 10 pink marbles. One marble is drawn at random from box. What is the probability that the marble drawn is either red or green?
Sol:
Total number of marbles= 8+6+10 =24, n=24
There are 6 green & 8 red marbles
Therefore, number of favorable cases=6+8=14
P (red or green) = 14/24=7/12

## Probability Formulas

Here are some basic probability formulas that are frequently used in quantitative aptitude exams, along with a table for quick reference:

Note: P(A) represents the probability of event A. P(B|A) represents the probability of event B given that event A has occurred. P(A and B) represents the probability of both events A and B occurring.

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## FAQs on Overview: Probability - CAT

 1. What is the definition of probability?
Ans. Probability is a measure that quantifies the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 represents impossibility and 1 represents certainty.
 2. What is a random experiment?
Ans. A random experiment is a process or activity that can result in different outcomes, and the outcome cannot be predicted with certainty. Examples of random experiments include flipping a coin, rolling a die, or selecting a card from a deck.
 3. What are mutually exclusive events?
Ans. Mutually exclusive events are events that cannot occur at the same time. In other words, if one event happens, the other event cannot happen. For example, when rolling a die, the events of getting an odd number and getting an even number are mutually exclusive.
 4. What are equally likely events?
Ans. Equally likely events are events that have the same probability of occurring. For example, when rolling a fair six-sided die, each number (1, 2, 3, 4, 5, 6) has an equal chance of appearing, making them equally likely events.
 5. What are independent events?
Ans. Independent events are events where the occurrence or non-occurrence of one event does not affect the probability of the other event. In other words, the outcome of one event has no influence on the outcome of the other event. For example, if you toss a fair coin twice, the outcome of the first toss does not affect the outcome of the second toss, making them independent events.
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