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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
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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
2
http://www.shodor.org/interactivate/activities/ShapeSorter/
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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/
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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
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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/
