PROBABILITY Probability Concepts

Theoretical Probability vs Relative

Frequency

Calculating Probabilities

Venn Diagrams

Intersection of Sets

Union of Sets

Mutually Exclusive Events

Inclusive Events

Complementary Events

Probability Games in Life 1

VENN DIAGRAMS

Venn diagrams represent the sample space.

Example

Consider the hat experiment where

S = {1,2,3,4,5,6,7,8,9,10,11,12}.

Suppose that there are two events:

A = {drawing numbers less than or equal to 6}

= {1,2,3,4,5,6}

B = {drawing numbers greater than 6}

= {7,8,9,10,11,12}

Represent this on a Venn Diagram:

Basic Venn Diagrams

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http://www.shodor.org/interactivate/activities/ShapeSorter/

INTERSECTION OF SETS

The intersection occurs where the elements share a

common space. These are called inclusive events.

Example

Consider the hat experiment

where S = {l, 2,3,4,5,6,7,8,9,10,11,12}.

There are 2 events:

C = {drawing a factor of 6} = {1,2,3,6}

D = {drawing a factor of 9} = {l, 3,9}

Find the elements that intersect.

C D={l,3}

C and D ={1,3}

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UNION OF SETS

The union of A and B is an event consisting of all outcomes

that are in A or B.

Example

Determine the union of C and D.

C D ={l,2,3,6,9}

C or D ={1,2,3,6,9}

Here the numbers 4,5,7,8,10,11,12 are excluded from the

union of C and D. The number 1 and 3 appear in both set

C and D and are written only once in the union set.

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MUTUALLY EXCLUSIVE EVENTS

Events with no elements in common. Event A and B

exclude each other. If A happens, then B cannot happen.

Both cannot happen at the same time.

Example

a) Find the intersection of A and B:

A B = { }

A and B = { } empty set

P(A B) = 0

b) Find the union of A and B:

A B = {1;2;3;4;5;6;7;8;9;10;11;12}

A or B = {1;2;3;4;5;6;7;8;9;10;11;12}

P(A B) = P(A) + P(B) – P( A B)

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INCLUSIVE EVENTS

Events with elements in common.

Example

a) Find the intersection of C and D:

C D = {1;3}

C and D = {1;3 }

b) Find the union of C and D:

C D = {1;2;3;6;9}

C or D = {1;2;3;6;9}

P(C D) = P(A) + P(B) - P(A B)

Practicing Venn

Diagrams

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http://www.shodor.org/interactivate/activities/VennDiagrams/http://www.shodor.org/interactivate/activities/VennDiagrams/

COMPLEMENTARY EVENTS

If Event A and Event B is mutually exclusive, then Event A

and Event B are complementary.

Example

a) Find the complement of A.

A = {1;2;3;4;5;6}

Complement of A = Not A (A’)

= B = {7;8;9;10;11;12}

P(A) + P(A')= 1

b) Find the complement of B.

B = {7;8;9;10;11;12}

Complement of B = Not B (B’)

= A = {1;2;3;4;5;6}

P(B) + P(B')= 1 …. P (not B) = 1 - P(B)

Playing Cards &

Venn Diagrams 7

http://www.khanacademy.org/math/probability/v/probability-with-playing-cards-and-venn-diagramshttp://www.khanacademy.org/math/probability/v/probability-with-playing-cards-and-venn-diagrams

EXERCISE

Cards numbered from 1 to 12 are put into a box and shaken. Cards are then drawn and replaced. The following events are given:

A = {drawing an even number}

B = {drawing an odd number}

C = {drawing a number greater than 7}

D = {drawing a number less than 5}

E = {drawing natural numbers less than 7}

F = {drawing natural numbers greater than 4}

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(a) Draw a Venn Diagram to represent events A and B.

(b) Determine P(A B)

(c) Determine P(A B)

(d) Show that A and B are mutually exclusive.

(e) Are events A and B complementary?

(f) Draw a Venn Diagram to represent events A and C.

(g) Determine P(A or C)

(h) Determine P(A and C)

(i) Show that A and C are inclusive.

(j) Are events A and C complementary?

(k) Draw a Venn Diagram to represent events C and D.

(1) Determine whether C and D are mutually exclusive or

inclusive; complementary or not complementary.

(m) Draw a Venn Diagram to represent events E and F.

(n) Determine whether E and F are mutually exclusive or

inclusive; complementary or not complementary.

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Conditional Probability: Pick the Correct Door!

Picking Cards or Rolling Die

PROBABILITY GAMES IN LIFE

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http://www.shodor.org/interactivate/activities/AdvancedMontyHall/http://www.shodor.org/interactivate/activities/CrazyChoicesGame/