10.Probability

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Complete Study Guide & Notes On

PROBABILITY-THEORY OF CHANCES

One can’t just sit and fear Mathematics. One has to have some courage. In fact courage is not the acknowledgement of our fears. Rather it is the acknowledgement of something

which is more important than our fears!

IMPORTANT TERMS, DEFINITIONS & RESULTS

01. Basics Of Probability: Let S and E be the sample space and an event in an experiment respectively.

Then, ENumber of favourable events

Probability =Total number of elementary events S

n

n .

Also as E S

E Sn n n or 0 E Sn n

E S0

S S S

n n

n n n 0 P E 1 .

Hence, if P E denotes the probability of occurrence of an event E then, 0 P E 1 and

P E = 1 P E such that P E denotes the probability of non-occurrence of the event E.

Note that P E can also be represented as P E .

02. Mutually Exclusive or Disjoint Events: Two events A and B are said to be mutually exclusive if occurrence of one prevents the occurrence of the other i.e., they can’t occur simultaneously.

In this case, sets A and B are disjoint i.e., AB .

Consider an example of throwing a die. We have the sample spaces as, S 1,2,3,4,5,6 .

Suppose A 5,6the event of occurence of a number greater than 4 ,

B 1,3,5the event of occurence of an odd number and

C 2,4,6the event of occurence of an even number .

In these events, the events B and C are mutually exclusive events but A and B are not mutually exclusive events because they can occur together (when the number 5 comes up). Similarly A and C are not mutually exclusive events as they can also occur together (when the number 6 comes up).

If A and B are mutually exhaustive events then we always have,

P A B 0 As A B ( ) 0n n

P A B P A P B .

If A, B and C are mutually exhaustive events then we always have,

P A B C P A P B P C .

03. Independent Events: Two events are independent if the occurrence of one does not affect the occurrence of the other. Consider an example of drawing two balls one by one with replacement from a bag containing 3 red and 2 black balls. Suppose A the event of occurence of a red ball in first draw ,

B the event of occurence of a black ball in the second draw .

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RECAPITULATION & CONDITIONAL PROBABILITY

Probability By OP Gupta (INDIRA AWARD Winner, Elect. & Comm. Engineering)


Then 3

P A5

, 2

P B5

.

Here probability of occurrence of event B is not affected by occurrence or non-occurrence of the event A. Hence event A and B are independent events.

But if the two balls would have been drawn one by one without replacement, then the probability of

occurrence of a black ball in second draw when a red ball has been drawn in first draw 2

P B4

. Also

if a red ball is not drawn in the first draw, then the probability of occurrence of a black ball in the second

draw 1

P B4

(After a black ball is drawn there are only 4 balls left in the bag). In this case the event

of drawing a red ball in the first draw and the event of drawing a black ball in the second draw are not independent.

For independent events A and B, we always have P A B P A .P B .

For independent events A, B and C, we always have P A B C P A .P B .P C .

Also we have for independent events A and B, P A B 1 P A .P B .

Since for independent events A and B we have P A B P A .P B , so the conditional

probability (discussed later in this chapter) of event A when B has already occurred is given as,

P A BP A|B

P B

P A .P B

i.e., P A|BP B

P A|B P A .

04. Exhaustive Events: Two or more events say A, B and C of an experiment are said to be exhaustive events if, a) their union is the total sample space i.e. A B C S b) the events A, B and C are disjoint in pairs i.e. A B , B C and C A .

c) P A P B P C 1 .

Consider an example of throwing a die. We have S 1,2,3,4,5,6 .

Suppose A 2,4,6the event of occurence of an even number ,

B 1,3,5the event of occurence of an odd number and

C 3,6the event of getting a number multiple of 3 .

In these events, the events A and B are exhaustive events as A B S but the events A and C or the events B and C are not exhaustive events as A C S and similarly B C S .

05. Conditional Probability: By the conditional probability we mean the probability of occurrence of event A when B has already occurred. You can note that in case of occurrence of event A when B has already occurred, the event B acts as the sample space and A B acts as the favourable event. The ‘conditional probability of occurrence of event A when B has already occurred’ is sometimes also called as probability of occurrence of event A w.r.t. B.

P A B

P A | B , B . . P B 0P B

i e

P A B

P B | A , A . . P A 0P A

i e

P A B

P A | B , P B 0P B

P A B

P A | B , P B 0P B

P A B

P A | B , P B 0P B

P A | B P A | B 1, B .

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06. Useful formulae:

a) P A B P A P B P A B . . P A or B P A P B P A and Bi e

b) P(A B C) P(A) P(B) P(C) P(A B) P(B C) P(C A) P(A B C)

c) P A B P only B P B A P B but not A P B P A B

d) P A B P only A P A B P A but not B P A P A B

e) P A B P neither A nor B 1 P A B

!C , C C such that

! !n n n

r r n r

nn r

r n r

! .( 1).( 2)...5.4.3.2.1n n n n . Also, 0! 1 .

07. Events and Symbolic representations:

Verbal description of the event Equivalent set notation Event A A

Not A A or A

A or B (occurrence of atleast one of A and B) A B or A B

A and B (simultaneous occurrence of both A and B) A B or A B

A but not B ( A occurs but Bdoes not) A B or A B Neither A nor B A B At least one of A, B or C A B C All the three of A, B and C A B C

Description Of The Playing Cards:

Total no. of playing cards : 52

Total no. of Red cards : 26 & Total no. of Black cards : 26 (so, 26 + 26 = 52)

Four Suits are : Club, Spade (Both black); Heart, Diamond (Both red)

No. of cards in each suit : 13 (so, 13 4 = 52)

No. of Ace cards : 4 (2 Black & 2 Red)

Three Face cards are : Jack, Queen, King

No. of Face cards : 4 Jacks, 4 Queens, 4 Kings (so, total face cards : 4 3 = 12)

No. of Red Coloured Face cards : 6 (Two from each of the faces)

No. of Black coloured Face cards) : 6 (Two from each of the faces)

Numbering on the cards is done from 2 to 9. Sometimes Ace is numbered as 1 but if so, then it will be mentioned in the question. (so, we have four cards with each of the numbers from 2 to 9 from each of the four suits.)


01. Bayes’ Theorem: If 1 2 3E , E , E ,..., En are n non- empty events constituting a partition of sample

spaces S i.e., 1 2 3E , E , E ,..., En are pair wise disjoint and 1 2 3E E E ... E Sn and A is any event

of non-zero probability then,

1

P E .P A | EP E | A , 1,2,3,...,

P E .P A | E

i ii n

j jj

i n

TOTAL PROBABILITY & BAYES' THEOREM

Probability By OP Gupta (INDIRA AWARD Winner, Elect. & Comm. Engineering)


For example,

1 1

1

1 1 2 2 3 3

P E . P A | EP E |A

P E .P A | E P E .P A | E P E . P A | E

.

Bayes’ Theorem is also known as the formula for the probability of causes.

If 1 2 3E , E , E ,..., En form a partition of S and A be any event then,

1 1 2 2P A P E .P A | E P E . P A | E ... P E .P A | En n

P(E A) P(E ).P(A|E )i i i

The probabilities 1 2P E , P E ,...,P En which are known before the experiment takes place

are called priori probabilities and P A | En are called posteriori probabilities.


01. Bernoulli Trials: Trials of a random experiment are called Bernoulli trials, if they satisfy the following four conditions:

a) The trials should be finite in numbers. b) The trials should be independent of each other. c) Each of the trial yields exactly two outcomes i.e. success or failure. d) The probability of success or the failure remains the same in each of the trial.

02. Binomial Distribution: Let E be an event. Let p probability of success in one trial (i.e., occurrence

of event E in one trial) and, 1q p probability of failure in one trial (i.e.,non- occurrence of event E

in one trial). Let X = number of successes (i.e., number of times event E occurs in n trials). Then, Probability of X successes in n trials is given by the relation,

P(X ) P( ) Cn r n rrr r p q

where 0,1, 2,3,...,r n ; p probability of success in one trial and 1q p probability of failure in

one trial.

P(X )r or P( )r is also called probability of occurrence of event E exactly r times in n

trials.

Here !

C! ( )!

n

r

n

r n r

.

Note that Cn r n rr p q

is the ( 1) thr term in the binomial expansion of n

q p .

0

. P( )n

r

Mean r r np

22

0

.P( ) Meann

r

Variance r r npq

Standard Deviation npq

If an experiment is repeated n times under the similar conditions, we say that n trials of the experiment have been made.

The result P(X ) P( ) Cn r n rrr r p q can be used only when:

(i) the probability of success in each trial is the same. (ii) each trial must surely result in either a success or a failure.

BERNOULLI TRIALS & BINOMIAL DISTRIBUTION

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, P( 1) P( )1

n r pRecurrence formula x r + r

r q

.

A Binomial Distribution with n Bernoulli trials and probability of success in each trial as p is

denoted by B ,n p . Here n and p are known as the parameters of binomial distribution.

The expression P x r or P r is called the probability function of the binomial

distribution.


01. Random Variable: A random variable is a real valued function defined over the sample space of an experiment. In other words, a random variable is a real-valued function whose domain is the sample space of a random experiment. A random variable is usually denoted by upper case letters X, Y, Z etc.

Discrete random variable: It is a random variable which can take only finite or countably infinite number of values. Continuous random variable: It is a random variable which can take any value between two given limits is called a continuous random variable.

02. Probability Distribution Of A Random Variable: If the values of a random variable together with the corresponding probabilities are given, then this description is called a probability distribution of the random variable.

Mean or Expectation of a random variable 1

Pn

i ii

X x

Variance

2 2 2

1

( ) Pn

i ii

x

Standard Deviation Variance .

PROBABILITY DISTRIBUTION

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10.Probability