By Shivam, Sumit and Shivansh 11th

8/14/2019 By Shivam, Sumit and Shivansh 11th

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By shivam, sumit andshivansh

11th - D

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Topics

Introduction

Empirical Probability

Theoretical Probability

Compound Events

Addition Rule

Permutations

Combinations


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Probability Introduction

When we speak of the probability ofsomething happening, we are referring to

the likelihoodor chancesof it

happening. Do we have a better chance

of it occurring or do we have a better

chance of it not occurring?


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Generally, we talk about this probability

as a fraction, a decimal, or even a percent

the probability that if two dice are

tossed the spots will total to seven is 1/6

the probability that a baseball player

will get a hit is .273the probability that it will rain is 20%


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Empirical Probability

Some probabilities are determined fromrepeated experimentation and

observation, recording results, and then

using these results to predict expected

probability. This kind of probability is

referred to as empirical probability.


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If we conduct an experiment and record

the number of times a favorable event

occurs, then the probability of the event

occurring is given by:


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We can see this in the following

example. If we flip a coin 500 times and

it lands on heads 248 times, then the

empirical probability is given by:

248( ) 0.5500

P heads =

# of times event occurred( )

total # of times experiment performed

EP E =

Remember


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Theoretical Probability

Other probabilities are determined usingmathematical computations based on

possible results, or outcomes. This kind

of probability is referred to as theoretical

probability.


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The theoretical probability of eventEhappening is given by:

# of ways can occur( )

total # of possible outcomes

EP E =


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If we consider a fair coin has two sides

and only one side is heads, and either

side is likely to come up, then the

theoretical probability of tossing heads is

given by:

# of ways can occur( )

total # of possible outcomes

EP E =

Remember

# sides that are heads 1( ) 0.5

total number of sides 2P E = = =


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While in both cases illustrated for tossing

a heads the probability comes out to be

0.5, it should be noted that empirical

probability falls under the Law of Large

Numbers which basically says that anexperiment must be conducted a large

number of times in order to determine

the probability with any certainty.


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You can flip a coin ten times and have

heads come up seven times, but this does

not mean that the probability is 0.7. The

more times a coin is flipped, the more

certainty we have to determine theprobability of coming up heads.


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Other examples of theoretical probability

are found in determining the probability

of drawing a certain card from a standard

deck of cards.

A standard deck has four suits: spades(), hearts (), diamonds (), and clubs

(). It has thirteen cards in each suit:

ace, 2, 3, . . ., 10, jack, queen, and king.Each of these cards is equally likely to be

drawn.


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These examples lead to four rules or

facts about probability:

1. The probability of an event that

cannot occur is 0.

2. The probability of an event thatmust occur is 1.

3. Every probability is a number

between 0 and 1 inclusive.4. The sum of the probabilities of

all possible outcomes of an

experiment is 1.


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The complement of an event is all

outcomes where the desired event does

not occur. We can say the complement

ofEis notE(sometimes written asor

E).'


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Since any event will either occur or it

will not occur, by rule 4 previously

discussed, we get:

Remember

Rule 4: the sum of the

probabilities of all possible

outcomes of an experiment is 1.

( ) ( ) 1P E P not E + =


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can also be stated as:

So the probability of tossing a die and

not rolling a 4 is:

( ) ( ) 1P E P not E + =

1 5( 4) 1 (4) 1

6 6P not P = = =

( ) 1 ( )P not E P E =


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Compound Events

A compound event is an event consistingof two or more simple events. Examples

of simple events are: tossing a die and

rolling a 5, picking a seven from a deckof cards, or flipping a coin and having a

heads show up.


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An example of a compound event is

tossing a die and rolling a 5 or an even

number. The notation for this kind of

compound event is given by .

This is the probability that eventA oreventB (or both) will occur.

( or )P A B


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Addition Rule

This leads to the Addition Rule forcompound events. The statement of this

rule is that the probability of either of

two events occurring is the probabilityfor the first event occurring plus the

probability for the second event

occurring minus the probability of both

event occurring simultaneously.


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Stated mathematically the rule is given

by:

Thus, the probability of drawing a 3 or a

club from a standard deck of cards is:

4 13 1 16 4(3 or club)

52 52 52 52 13P = + = =

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

Cards with a 3

Cards with clubs Card that is a 3 and a club


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If two events are mutually exclusive,

they cannot occur simultaneously.

Therefore, , and the

Addition Rule for mutually exclusive

events is given by:

( or ) 0P A B =

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


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Permutations

A permutation is an arrangement ofobjects where order is important. For

instance the digits 1,2, and 3 can be

arranged in six different orders --- 123,132, 213, 231, 312, and 321. Hence,

there are six permutations of the three

digits. In fact there are six permutationsof any three objects when all three

objects are used.


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In general the number of permutations

can be derived from the Multiplication

Principal. For three objects, there are

three choices for selecting the first object.

Then there are two choices for selecting

the second object, and finally there is only

one choice for the final object. This gives

the number of permutations for threeobjects as 3 2 1=6.


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Now suppose that we have 10 objects

and wish to make arrangements by

selecting only 3 of those objects. For the

first object we have 10 choices. For the

second we have 9 choices, and for the

third we have 8 choices. So the number

of permutations when using 3 objects out

of a group of 10 objects is 10 9 8=720.


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We can use this example to help derive

the formula for computing the number of

permutations ofrobjects chosen from ndistinct objects r n. The notation forthese permutations is and the

formula is:

( , )P n r

( , ) ( 1) ( 2) ... [ ( 1)]P n r n n n n r =


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We often use factorial notation to rewrite

this formula. Recall that:

And

Using this notation we can rewrite the

Permutation Formula for as

! ( 1) ( 2) ( 3) ... 3 2 1n n n n n=

0! 1=

( , )P n r

!( , )

( )!

nP n r

n r=


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It is important to remember that in using

this formula to determine the number of

permutations:

1. The n objects must be distinct2. That once an object is used it

cannot be repeated

3. That the order of objects is

important.


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Combinations

A combination is an arrangement ofobjects in which order is not important.

We arrange robjects from among n

distinct objects where r n. We use thenotation C(n, r) to represent thiscombination. The formula for C(n, r) is

given by: !( , )

( )! !

nC n r

n r r=


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The Combination Formula is derived

from the Permutation Formula in that for

a permutation every different order of the

objects is counted even when the same

objects are involved. This means that for

robjects, there will be r! different orderarrangements.


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Permutation Combination

So in order to get the number of different

combinations, we must divide the

number of permutations by r!. The result

is the value we get for C(n, r) in theprevious formula.

!

( , ) ( )!

nP n r

n r=

!

( , ) ( )! !

nC n r

n r r=

By Shivam, Sumit and Shivansh 11th

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