Probability Mathematic

Probability:- Probability means the chances of happening/occurring of an event. So, in this chapter we discuss about the Probability of an event to happen/occur. We usually predict about many events based on certain parameters.

Probability can be used to predict the likelihood of an event occurring, and can be used to make decisions about how to best manage risk. Probability theory is a powerful tool that can be used to help us understand the world and make better decisions.

example, if you roll a dice 100 times and it lands on a 4 20 times, then the experimental probability of rolling a 4 is 20%.

For example :-

♦ Getting a head or tail, when a coin is tossed .

♦ Getting a number from 1 to 6, when a die is rolled .

The better we know about the parameters related to an event better will be the accuracy of the result predicted. Mathematically, we can say that probability of happening an event is equal to the ratio of number of favourable outcomes to number of possible outcomes. It is represented as shown below Probability happening of an event

P = Number of favourable Outcomes / Total Number of possible Outcomes

Terms Related to Probability :-

Various terms related to probability are as follows

Experiment

An action where the result is uncertain even though the all possible outcomes related to it is known in advance. This is also known as random experiment, e. g., Throwing a die, tossing a coin etc.

Sample Space

A sample space of an experiment is the set of all possible outcomes of that experiment. It is denoted by S. For example If we throw a die,

then sample space S = {1, 2, 3, 4, 5, 6} If we toss a coin, then sample space S = {Head, Tail}

Possible outcomes

All possibilities related to an event are known as possible outcomes.

Tossing a Coin When a coin is tossed, these are two possible outcomes.

So, we say that the probability of getting H is 1/2 or the probability of getting T is 1/2,

Throwing a Die When a single die is thrown, there are six possible outcomes 1, 2, 3, 4, 5 and 6

The probability of getting any one of this numbers is 1/6

Ex.1:-

There are 5 marbles in a bag. 3 of them are red and 2 of them are blue. What is the probability that a blue marble will be picked?

Answer :- Number of favourable outcomes = 2 (because there are 2 blue marbles) Total number of outcomes = 5 (because there are 5 marbles in total).

So required Probability is = 2/5 = 0.4

👉 Event

Event is the single result of an experiment, e. g., Getting a head is an event related to tossing of a coin.

Types of Events

Various types of events are as follows

Certain and Impossible Events

A certain event is certain to occur, i.e., S (sample space) is a certain event. Probability of certain event is 1, i.e., P[S) = 1.

An impossible event has no chance of occurring, i.e., § is the impossible event. Probability of impossible event is 0, i. e., P (0) = 0.

Ex. 2:- A teacher chooses a student at random from a class of 30 boys. What is the probability that the student chosen is a boy?

Answer:- since all the students are boys so, chosen may be any one ,

i.e. favourable cases = Total cases = 30

Probability = 30/30 = 1

Ex. 3:- A bag contains 20 black marbles, if a marble is picked at random from the bag. Find the probability that marble picked is of Red colour .

Answer:-. The bag contains 20 black marbles and there is no red marble in the bag

So favourable cases = 0 Total outcomes = 20

Required Probability = 0/20 = 0

Equally Likely Events

Events related to an experiment are said to be equally likely events,

if probability of occurrence of each event is same.

👉 For example When a dice is rolled the possible outcome of getting an odd number = possible out come of getting an even number = 3.

So getting a even number or odd number are equally likely events.

Complement of an Event

The complement of an event A is the set of all outcomes in the sample space that are not included in the outcomes of event A. The complement of event A is represented by A (read as A bar).

The probability of complement of an event can be found by subtractina the given probability from 1

Mutually Exclusive and Exhaustive Events

♦ Two events E± and E2 related to an experiment E, having sample space S are known as mutually exclusive, if the probability of occurrence of both events simultaneously is zero.

👉 i. e.,

P(E1nE2)=Q For example When a coin is tossed either head or tail will appear. Head and tail

con not occur simultareously. Therefore occurrence of a head or a tail are two mutually exclusive events.

■f Two events E1 and E2 related to an experiment E, having sample space S are known as mutually exhaustive, if the probability of occurrence of event E1 or E2 isl.

i.e., P{E1<uE2)=l

For example Let A be probability of getting an even number when a dice is rolled and B be the

probability of getting an odd number. The probability of occurrence of event A or event B is 1

i.e., any of the even can occur so they are mutually exhanstive. Note Events E1, E2, E3 ■■■En

related to Sare known as

1. Mutually exclusive, if P (Er n E2 n£3-n£n) = 0 orE: nE2 r\E3 ■•■En = 0

2. Mutually exhaustive, if P(E1 uE2 UE3 • • • <jEn) = 1 or Et u E2 u E3 ■■■En = S

Dependent Events

Two events are called dependent, if the outcome or occurrence of the first affects the outcome or occurrence of the second, so that the probability is changed.

Independent Events

Two events A and B are called independent, if occurring or non-occurring of A does not affect the occurring or non-occurring of B. If A and B are independent events,

then P (A and B) = P {AnB) = P(A)-P (B)

For example, getting head after tossing a coin and getting a 5 on a rolling single 6-sided die are independent events.