NCERT Class 11-Math's: Exemplar Chapter –16 Probability Part 1

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15.1 Overview

Probability is defined as a quantitative measure of uncertainty – a numerical value that conveys the strength of our belief in the occurrence of an event. The probability of an event is always a number between 0 and 1 both 0 and 1 inclusive. If an event’s probability is nearer to 1, the higher is the likelihood that the event will occur; the closer the event’s probability to 0, the smaller is the likelihood that the event will occur. If the event cannot occur, its probability is 0. If it must occur (i.e., its occurrence is certain), its probability is 1.

16.1.1 Random experiment:

An experiment is random means that the experiment has more than one possible outcome and it is not possible to predict with certainty which outcome that will be. For instance, in an experiment of tossing an ordinary coin, it can be predicted with certainty that the coin will land either heads up or tails up, but it is not known for sure whether heads or tails will occur. If a die is thrown once, any of the six numbers, i.e., 1, 2, 3, 4, 5, 6 may turn up, not sure which number will come up.

(i) Outcome: A possible result of a random experiment is called its outcome for example if the experiment consists of tossing a coin twice, some of the outcomes are HH, HT etc.

(ii) Sample Space: A sample space is the set of all possible outcomes of an experiment. In fact, it is the universal set S pertinent to a given experiment.

The sample space for the experiment of tossing a coin twice is given by

The sample space for the experiment of drawing a card out of a deck is the set of all cards in the deck.

16.1.2 Event:

An event is a subset of a sample space S. For example, the event of drawing an ace from a deck is

16.1.3 Types of events:

(i) Impossible and Sure Events: The empty set φ and the sample space S describe events. In fact φ is called an impossible event and S, i.e., the whole sample space is called a sure event.

(ii) Simple or Elementary Event: If an event E has only one sample point of a sample space, i.e., a single outcome of an experiment, it is called a simple or elementary event. The sample space of the experiment of tossing two coins is given by

The event containing a single outcome of the sample space S is a simple or elementary event. If one card is drawn from a well shuffled deck, any particular card drawn like ‘queen of Hearts’ is an elementary event

(iii) Compound Event If an event has more than one sample point it is called a compound event, for example, is a compound event.

(iv) Complementary event Given an event A, the complement of A is the event consisting of all sample space outcomes that do not correspond to the occurrence of A.

The complement of A is denoted by or . It is also called the event ‘not . Further denotes the probability that A will not occur.

16.1.4 Event ‘A or B’:

If A and B are two events associated with same sample space, then the event ‘A or B’ is same as the event and contains all those elements which are either in A or in B or in both. Further-more, denotes the probability that A or B (or both) will occur.

16.1.5 Event ‘A and B’:

If A and B are two events associated with a sample space, then the event ‘A and B’ is same as the event and contains all those elements which are common to both A and B. Further-more, denotes the probability that both A and B will simultaneously occur.

16.1.6 The Event ‘A but not B’ (Difference A – B):

An event A – B is the set of all those elements of the same space S which are in A but not in B, i.e., A – B = A ∩ B′.

16.1.7 Mutually exclusive:

Two events A and B of a sample space S are mutually exclusive if the occurrence of any one of them excludes the occurrence of the other event. Hence, the two events A and B cannot occur simultaneously, and thus .

Remark Simple or elementary events of a sample space are always mutually exclusive.

For example, the elementary events or of the experiment of throwing a dice are mutually exclusive.

Consider the experiment of throwing a die once.

The events getting a even number and getting an odd number are mutually exclusive events because .

Note: For a given sample space, there may be two or more mutually exclusive events.

16.1.8 Exhaustive events:

If are events of a sample space and if

then are called exhaustive events.

In other words, events E1, E2, ..., En of a sample space S are said to be exhaustive if atleast one of them necessarily occur whenever the experiment is performed.

Consider the example of rolling a die. We have . Define the two events number less than or equal to 4 appears.’

number greater than or equal to 4 appears.’

Now

Such events A and B are called exhaustive events.