{"id":281,"date":"2024-04-02T07:22:10","date_gmt":"2024-04-02T06:22:10","guid":{"rendered":"https:\/\/tutors4you.com\/?page_id=281"},"modified":"2024-04-02T07:22:57","modified_gmt":"2024-04-02T06:22:57","slug":"probability-tutorial","status":"publish","type":"page","link":"https:\/\/tutors4you.com\/index.php\/probability-tutorial\/","title":{"rendered":"Probability: basic concepts"},"content":{"rendered":"\n
\n\t\t\t\n\t\t\n

Random Experiment<\/h3> \n

An experiment is said to be a random experiment, if it’s out-come can’t be\n predicted with certainty.<\/p>\n \n

Example<\/h4> If a coin is tossed, we can’t say, whether head or tail will appear. So it is a\n random experiment.<\/p>\n

Sample Space<\/h3>\n

The set of all possible out-comes of an experiment is called the sample\n space. It is denoted by ‘S’ and its number of elements\n are n(s).<\/p>\n \n

Example<\/h4> In throwing a dice, the number that appears at top is any one of 1,2,3,4,5,6. So\n here:<\/p>\n \n

S ={1,2,3,4,5,6} and n(s) = 6<\/p>\n \n

Similarly in the case of a coin, S={Head,Tail} or {H,T} and n(s)=2.<\/p>\n \n

Elements<\/h3>\n\t\t

The elements of the sample space are called sample point or event point.<\/p>\n

Event<\/h3>\n

Every subset of a sample space is an event. It is denoted by ‘E’.<\/p>\n \n

Example<\/h4> In throwing a dice S={1,2,3,4,5,6}, the appearance of an event number will\n be the event E={2,4,6}.<\/p>\n \n

Clearly E is a sub set of S.<\/p>\n

Simple event<\/h3>\n

An event, consisting of a single sample point is called a simple event.<\/p>\n \n

Example<\/h4> In throwing a dice, S={1,2,3,4,5,6}, so each of {1},{2},{3},{4},{5} and {6}\n are simple events.<\/p>\n

Compound event<\/h3>\n

A subset of the sample space, which has more than on element is called a mixed event.<\/p>\n \n

Example<\/h4> In throwing a dice, the event of appearing of odd numbers is a compound event,\n because E={1,3,5} which has ‘3’ elements.<\/p>\n

Equally likely events<\/h3>\n

Events are said to be equally likely, if we have no reason to believe that one is more likely to occur than the other.<\/p>\n \n

Example<\/h4> When a dice is thrown, all the six faces {1,2,3,4,5,6} are equally likely to come up.<\/p>\n

Exhaustive events<\/h3> \n

When every possible out come of an experiment is considered.<\/p>\n \n

Example<\/h4> A dice is thrown, cases 1,2,3,4,5,6 form an exhaustive set of events.<\/p>\n

Classical definition of probability:<\/h3> \n \n

If ‘S’ be the sample space, then the probability of occurrence of an event ‘E’ is defined\n as:<\/p>\n \n <\/br><\/br>\n \n

Example<\/h4> Find the probability of getting a tail in tossing of a coin.<\/p>\n \n

Solution<\/h4>\n \n

Sample space S = {H,T} and\n n(s) = 2<\/p>\n \n

Event ‘E’ = {T} and n(E) = 1<\/p>\n \n

therefore <\/br><\/br><\/p>\n \n

Note: This definition is not true, if
\n (a) The events are not equally likely.
\n (b) The possible outcomes are infinite.<\/p>\n

Sure event<\/h3>\n

Let ‘S’ be a sample space. If E is a subset of or equal to S then E is called a sure event.<\/p>\n \n

Example<\/h4> In a throw of a dice, S={1,2,3,4,5,6}<\/p>\n \n

Let E1<\/sub>=Event of getting a number less than ‘7’.<\/p>\n \n

So ‘E1<\/sub>‘ is a sure event.<\/p>\n \n

So, we can say, in a sure event n(E) = n(S)<\/p>\n

Mutually exclusive or disjoint event<\/h3> \n

If two or more events can’t occur simultaneously, that is no two of them can occur together.<\/p>\n \n

Example<\/h4> When a coin is tossed, the event of occurrence of a head and the event of\n occurrence of a tail are mutually exclusive events.<\/p>\n\t\t\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t

Pictorial Representation:<\/p>\n\t\t\t\t

<\/p>\n\t\t\t\t<\/p>\n\t\t\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n

Independent or mutually independent events<\/h3> \n

Two or more events are said to be\n independent if occurrence or non-occurrence of any of\n them does not affect the probability of occurrence or\n non-occurrence of the other event.<\/p>\n \n

Example<\/h4> When a coin is tossed\n twice, the event of occurrence of head in the first throw\n and the event of occurrence of head in the second throw\n are independent events.<\/p>\n

Difference between mutually exclusive an mutually independent events<\/h3>\n

Mutually exclusiveness id used when the events are taken from the\n same experiment, where as independence is used when the\n events are taken from different experiments.<\/p>\n

Complement of an event<\/h3> \n

Let ‘S’ be the sample for random experiment, and ‘E’ be an event, then\n complement of ‘E’ is is denoted by E1<\/sup>.\n Here E occurs, if and only if E1<\/sup> doesn’t occur.<\/p>\n \n\t\t\t\t\n\t\t\t <\/p>\n\t\t\t\t

Clearly <\/p>\n \n

Random Experiment<\/h3>\n

An experiment is said to be a random\n experiment, if its out-come can’t be predicted with\n certainly.<\/p>\n \n

Example<\/h4> If a coin is tossed, we can’t\n say, whether head or rail will appear. So it is a random\n experiment.<\/p>\n \n

Sample Space<\/h3> \n

The set of all possible out-comes of an\n experiment is called the sample – space.<\/p>\n \n

If a dice is thrown, the number, that\n appears at top is any one of 1, 2, 3, 4, 5, 6,<\/p>\n \n

So here :<\/p>\n \n \n

S = { 1, 2, 3, 4, 5, 6, } and n(s) = 6<\/p>\n \n

Similarly in the case of a coin, s = {H,T} and n (s) = 2.<\/p>\n \n

The elements of the sample of the\n sample-space are called sample points or event points.<\/p>\n \n

Example<\/h4> if S = {H, T}, than ‘H’\n and ‘T’ are sample points.<\/p>\n \n

Event: Every\n subset of a sample space is an event. It is denoted by\n ‘E’.<\/p>\n \n

Example<\/h4> In throwing a dice S = {1, 2, 3, 4, 5, 6,}, the appearance of an even number will be the\n event E = {2, 4, 6}.<\/p>\n \n

Clearly <\/p>\n \n

Important types of Events<\/h2>\n \n

Simple or elementary event<\/h3>\n

An event, consisting of a single point is\n called a simple event.<\/p>\n \n

Example<\/h4> In throwing a dice s = {1, 2,\n 3, 4, 5, 6} so each of {1}, {2}, {3}, {4}, {5} and {6} is\n a simple event.<\/p>\n \n

Compound or mixed event<\/h3> \n

A subset of the sample space which has\n more than one element is called a mixed event.<\/p>\n \n

Example<\/h4> In throwing a dice, the event of odd\n numbers appearing is a mixed event, because E = {1,\n 3, 5}, which has ‘3’ elements.<\/p>\n \n

Equally likely events<\/h3>\n

Events are said to be equally likely, if\n we have no reason to believe that one is more likely to\n occur than the other.<\/p>\n \n

Example<\/h4> When a dice is thrown, all the\n six-faces {1, 2, 3, 4, 5, 6,} are equally likely to\n come-up.<\/p>\n \n

Exhaustive events<\/h3>\n

When every possible outcome of an\n experiment is considered, the observation is called\n exhaustive events.<\/p>\n \n

Example<\/h4> When a dice is thrown, cases 1, 2,\n 3, 4, 5, 5 form an exhaustive set of events.<\/p>\n \n \n\t\t\n\t\t\n\t\t