addition and multi. theorum

7/31/2019 Addition and Multi. Theorum

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ADDITION AND

MULTIPLICATION THEOREMSOF PROBABILITY

ByPRERANA JHUNJHUNWALAMBA 1STSEMROLLNO. 26FMS, BHU


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WHAT IS PROBABILITY?

A probabilityis a measure of the likelihood that anevent in the future will happen. It can only assume avalue between0and1.

A value near 0 means the event is not likely tohappen. A value near 1 means it is likely.

ExampleP(A) = Number of favourable outcomes

Total number of possible outcomes


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TERMINOLOGIES

EXPERIMENT

a process of obtaining information through

observation or measurement of a phenomenonwhose outcome is subject to chance.

EXAMPLE:

Tossing a fair coin.

Drawing a card from the pack of 52 cards.

Contd


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OUTCOME

the particular result of an experiment.

EXAMPLE:

On tossing a coin, the outcome is either a heador a tail.

Contd


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EVENT

is the collection of one or more outcomes of an

experiment.

EXAMPLE:

throwing a die and getting a six is an event.

Contd


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SAMPLE SPACE

the set of all possible outcomes of an

experiment.

EXAMPLE:

Outcomes of the roll of a die: {1, 2, 3, 4, 5, 6}

Outcomes of 2 coin flips: {HH, HT, TH, TT}


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TYPES OF EVENTS

MUTUALLY EXCLUSIVE EVENTS

Events aremutually exclusiveif the occurrence ofany one event means that none of the others canoccur at the same time.

Events which cannot occur together orsimultaneously.

EXAMPLE:if a single coin is tossed, head can be up or tailcan be up, both cannot be up at the same time.

Contd


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INDEPENDENT EVENTS

Events areindependentif the occurrence ofone event does not affect the occurrence of

another.

EXAMPLE:

the outcomes of successive tosses of a coin

are independent of its preceding toss.

Contd


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DEPENDENT EVENTS

Events aredependentif the occurrence of one

event does affect the occurrence of another.

EXAMPLE

drawing of two cards from a pack of 52 cards

without replacement.

Contd


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COMPLEMENTARY EVENTSIf A is an event of a sample space, then itscomplement contains all the elements of thesample space that are not part of A. If S denotesthe sample space then

~A = S A

EXAMPLEon throwing a die, a face with number 1 comesup. Lets denote it as E.

then, ~E = S E

i.e., = 6 1 = 5


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RULES OF PROBABILITY

RULES OF PROBABILITY

MULTIPLICATION RULEADDITION RULE

INDEPENDENT EVENTS

MUTUALLY EXCLUSIVE

EVENTSDEPENDENT EVENTS

PARTIALLY OVERLAPPINGEVENTS


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THE ADDITION RULE

Mutually exclusive events

If two events A and B are mutuallyexclusive, the probability of one orthe other events occurring equals

the sum of their probabilities.

P(A or B) = P(A) + P(B)

Partially overlapping eventsIf A and B are two events that are notmutually exclusive, then P(A or B) is:P(A or B) = P(A) + P(B) - P(A and B)


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EXAMPLE 1 A survey of 200 grocery shops revealed following monthly income

patterns:

No. of ShopsMonthly Income (Rs.)

A. Under Rs. 20,000 102

B. 20,000 to 30,000 61

C. 30,000 and above 37

(a)What is the probability that a particular shop has monthly income underRs.20,000?

(b) What is the probability that a shop selected at random has either anincome between Rs.20,000 and Rs.30,000 or an income of Rs.30,000

and above?


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Contd

(a). P(A) = A/S

= 102/200

= 0.51

(b). P(A or B) = P(A) + P(B)

= 61/200 + 37/200= 0.305 + 0.185

= 0.49


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EXAMPLE 2

What is the probability that a card chosen at random

from a standard deck of cards will be either a king or

a heart?

P(A or B) = P(A) + P(B) - P(A and B)

= 4/52 + 13/52 - 1/52

= 16/52, or .3077


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THE MULTIPLICATION RULE

Independent events

If two events are independent, then the probability ofhappening of one and the other event equals theproduct of their probabilities.

P(A and B) = P(A).P(B)

Dependent events

The probability of two dependent events A and B

equals the probability of A multiplied by theprobability of B given that A has occurred.

P(A and B) = P(A).P(B/A)


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EXAMPLE 1

An university has to select an examiner from a list of50 persons, 20 of them women and 30 men, 10 ofthem knowing Hindi and 40 not, 15 of them beingteachers and the remaining 35 not. What is theprobability of the University selecting a Hindiknowing woman teacher?

Probability of selecting awoman=20/50Probability of selecting ateacher=15/50

Probability of selecting aHindi-knowing=10/50Since the events areindependent, the probability ofthe University selecting a Hindi- knowing womanteacher = 20/50.15/50. 10/50 = 3/125 =0.024


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EXAMPLE 2

A bag contains 5white and 3black balls. Two ballsare drawn at random one after the other withoutreplacement. Find the probability that both ballsdrawn are black.

Probability of drawing a black ball in the first attemptis P(A) =3/8Probability of drawing the second black ball giventhat the first ball drawn is black is P(B/A) =2/7

Therefore, the probability that both balls drawn areblack is P(AB) = P(A).P(B/A) = 3/8. 2/7 =3/28


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Some more numericalproblems A bag contains 30 balls numbered from 1 to 30. One

ball is drawn at random. Find the probability that thenumber of the balls drawn will be a multiple of

i). 5 or 7

ii). 3 or 7

i). Probability of the number being a multiple of 5 isP(5,10,15,20,25,30) =6/30Probability of the number being a multiple of 7 is

P(7,14,21,28) = 4/30

Since the events are mutually exclusive, theprobability of the number being a multiple of 5 or 7will be 6/30 + 4/30 = 10/30 =1/3

contd


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ii). Probability of the number being a multiple of 3 isP(3,6,9,12,15,18,21,24,27,30) =10/30

Probability of the number being a multiple of 7 isP(7,14,21,28) =4/30

Since 21 is a common multiple of both, hence theprobability of getting a number which is multiple of 3or 7 = 10/30 +4/30 1/30 =13/30.

{P(A or B) = P(A) + P(B) P(A and B)}


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A box contains 3red and 7white balls. One ball isdrawn at random and in its place a ball of the othercolour is put in the box . Now one ball is drawn atrandom from the box. Find the probability that it isred.

When the first ball drawn is redP(R) =3/10

Now a white ball is put in the box in place of red ball,

so the box contains 2red and 8white balls.So the probability of drawing a red ball again is

P(R) =2/10


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When the first ball drawn is whiteP(W) =7/10

Now a red ball is put in the box in place of white ball,so the box contains 4red and 6white balls.

So the probability of drawing a red ball again isP(R) =4/10

Hence the probability of drawing a red ball is(3/10.2/10) + (7/10.4/10)= 6/100 + 28/100= 34/100=0.34


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addition and multi. theorum

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