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Welcome to MM207
Unit 3 Seminar
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Probability
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Probability Experiment
Outcomes
Events
Sample Space
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Probability
• Experiment: Rolling a single die
• Sample Space: All possible outcomes from
experiment
S = {1, 2, 3, 4, 5, 6}
• Event: a collection of one or more outcomes
(denoted by capital letter)
Event A = {3}
Event B = {even number}
• Probability = (number of favorable outcomes) /
(total number of outcomes)
– P(A) = 1/6
– P(B) = 3/6 = ½
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Probability
• Probability will always be between 0 and 1.
It will never be negative or greater than 1.
• Complement of an event: All outcomes
that are not included in the Event of
interest.
– If A = {3} then the “not A”or A’ = {1, 2, 4, 5, 6}.
A’ is everything but 3
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Independent Events
• Two events are independent if the occurrence of one
does not affect the probability of the occurrence of the
other.
– Independent: flip a coin twice, record the result each time
– Dependent: Select one card from a deck, record result; select
a second card from the same deck, record result
Conditional Probability
• The probability of an event occurring, given that
another event has already occurred.
• P(B|A)
P(B given that A has occurred)
• If A and B are independent: P(B|A) = P(B)
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Probability Rules
• Multiplication Rule (Joint Probabilities)
The probability that two events will occur in sequence:
• P(A and B) = P(A) * P(B|A)
If the events are Independent events:
• P(A) * P(B)
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Mutually Exclusive Events
(Sometimes called Disjoint Events)
• If two events cannot occur at the same time then they
are mutually exclusive, or disjoint
– Mutually exclusive:
A = my favorite color is red, B=my favorite color is blue
– Not Mutually Exclusive:
A = I like the color red, B = I like the color blue
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Probability Rules
• Addition Rule
– Mutually Exclusive
P(A or B) = P(A) + P(B)
– Not Mutually Exclusive
P(A or B) = P(A) + P(B) – P(A and B)
Note: “or” is interpreted as A or B or both
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Permutations and Combinations
• Permutation: Arranging “n” distinct objects taken “r” at a time
– A, B, C is not the same as C, B, A (ORDER is important)
Example: Given the letters A, B, and C, what are the total number of permutations of
2 letters?
(how many ways can you arrange 2 of the 3 letters, order matters)
AB, BA, AC, CA, BC, CB
6 total
• Combination: Arranging “n” distinct objects taken “r” at a time but:
– 1, 2, 3, is the same as 3, 2, 1 (ORDER does NOT matter)
What are the total number of combinations of 2 letters? (How many ways can you
choose 2 letters, order doesn’t matter)
AB, AC, BC
3 total
Permutations and Combinations
Factorial notation: n! = n *(n-1)*(n-2)*….*3*2*1
Example: 5! = 5*4*3*2*1 = 120
Note: 1! = 1 and 0! = 1
Permutation: Arranging “n” distinct objects taken “r” at a time (order matters)
n!
Pn,r 
(n  r )!
Combination: Choosing“n” distinct objects taken “r” at a time (order does not matter)
C n,r
n!
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r!(n  r )!
Insert Function
Click on the fx
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Insert Function Dialog Box
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Excel Function for Permutations
Problem 7, Page 178
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Excel Function for Combinations
Problem 9, Page 178
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