learn Bcom probabilty

1. Outline
Addition and Multiplication Rules for Probability
Lecture 10, STAT 2246
Julien Dompierre
D´partement de math´matiques et d’informatique
e e
Universit´ Laurentienne
e
30 janvier 2007, Sudbury
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2. Addition Rules
Outline
Multiplication Rules
Outline
1 Addition and Multiplication Rules for Probability
Addition Rules
Multiplication Rules
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3. Addition Rules
Outline
Multiplication Rules
Outline
1 Addition and Multiplication Rules for Probability
Addition Rules
Multiplication Rules
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4. Addition Rules
Outline
Multiplication Rules
Mutually Exclusive Events (p. 195)
Two events of the same experiment are mutually exclusive
events if they cannot occur at the same time (i.e., they have no
outcomes in common).
U
A B
In this case, the intersection of the sets A and B is empty.
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5. Addition Rules
Outline
Multiplication Rules
Mutually Exclusive Events
If A and B are mutually exclusive events of the same experiment,
then the probability that A and B will occur is
n(A ∩ B) 0
P(A ∩ B) = = = 0.
n(S) n(S)
For example. The experiment is to roll a die. The sample space is
the set of all possible outcomes is S = {1, 2, 3, 4, 5, 6}.
The event A is to get an odd number, A = {1, 3, 5} ⊆ S.
The event B is to get a 6, B = {6} ⊆ S.
In probability theory, we say that the events A and B are mutually
exclusive because they have no outcomes in common.
In set theory, we say that the sets A and B are mutually exclusive
because their intersection is empty.
n(A ∩ B) n(∅) 0
P(A ∩ B) = = = = 0.
n(S) n(S) 6
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6. Addition Rules
Outline
Multiplication Rules
Addition Rule 1 (p. 196)
When two events A and B of the same experiment are mutually
exclusive, the probability that A or B will occur is
n(A ∪ B) n(A) n(B)
P(A ∪ B) = = + = P(A) + P(B).
n(S) n(S) n(S)
For example. The experiment is to roll a die. The sample space is
the set of all possible outcomes is S = {1, 2, 3, 4, 5, 6}.
The event A is to get an odd number, A = {1, 3, 5} ⊆ S.
The event B is to get a 6, B = {6} ⊆ S.
n(A ∪ B) n(A) n(B) 3 1
P(A ∪ B) = = + = + = 4/6.
n(S) n(S) n(S) 6 6
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7. Addition Rules
Outline
Multiplication Rules
Principle of Inclusion-Exclusion (p. 197)
When two events are not mutually exclusive, we must subtract one
of the two probabilities of the outcomes that are common to both
events, since they have been counted twice.
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
A B
A∩B
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8. Addition Rules
Outline
Multiplication Rules
Addition Rule 2 (p. 197)
When two events A and B of the same experiment are not
mutually exclusive, the probability that A or B will occur is
n(A ∪ B) n(A) + n(B) − n(A ∩ B)
P(A ∪ B) = =
n(S) n(S)
= P(A) + P(B) − P(A ∩ B).
Note: This rule can also be used when the events are mutually
exclusive, since (A ∩ B) will always equal 0. However, it is
important to make a distinction between the two situations.
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9. Addition Rules
Outline
Multiplication Rules
Example of Addition Rule 2
For example. The experiment is to roll a die. The sample space is
the set of all possible outcomes is S = {1, 2, 3, 4, 5, 6}.
The event A is to get an odd number, A = {1, 3, 5} ⊆ S.
The event B is to get a number greater than 4, B = {5, 6} ⊆ S.
As A ∩ B = {1, 3, 5} ∩ {5, 6} = {5} = ∅, the events A and B are
not mutually exclusive.
3 2 1 4
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = + − =
6 6 6 6
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10. Addition Rules
Outline
Multiplication Rules
Principe of Inclusion-Exclusion for Three Sets
n(A ∪ B ∪ C ) = n(A) + n(B) + n(C )
− n(A ∩ B) − n(A ∩ C ) − n(B ∩ C )
+ n(A ∩ B ∩ C ).
A A∩B B
A∩B ∩C
A∩C B ∩C
C
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11. Addition Rules
Outline
Multiplication Rules
Principe of Inclusion-Exclusion for Four Sets
n(A1 ∪ A2 ∪ A3 ∪ A4 )
= n(A1 ) + n(A2 ) + n(A3 ) + n(A4 )
− n(A1 ∩ A2 ) − n(A1 ∩ A3 ) − n(A1 ∩ A4 ) − n(A2 ∩ A3 )
− n(A2 ∩ A4 ) − n(A3 ∩ A4 )
+ n(A1 ∩ A2 ∩ A3 ) + n(A1 ∩ A2 ∩ A4 ) + n(A1 ∩ A3 ∩ A4 )
+ n(A2 ∩ A3 ∩ A4 )
− n(A1 ∩ A2 ∩ A3 ∩ A4 )
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12. Addition Rules
Outline
Multiplication Rules
Principe of Inclusion-Exclusion for n Sets
Let A1 , A2 , ..., An be n finite sets. Then
n(A1 ∪ A2 ∪ · · · ∪ An ) = n(Ai )
1≤i≤n
− n(Ai ∩ Aj )
1≤i<j≤n
+ n(Ai ∩ Aj ∩ Ak )
1≤i<j<k≤n
− ··· + ··· − ···
n+1
+ (−1) n(A1 ∩ A2 ∩ · · · ∩ An )
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13. Addition Rules
Outline
Multiplication Rules
Outline
1 Addition and Multiplication Rules for Probability
Addition Rules
Multiplication Rules
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14. Addition Rules
Outline
Multiplication Rules
Independent Events (p. 205)
The multiplication rules can be used to find the probability of two
or more events that occur in sequence. For example, if a coin
is tossed and then a die is rolled, one can find the probability of
getting a head on the coin and a 4 on the die. These two events
are said to be independent since the outcome of the first event
(tossing a coin) does not affect the probability outcome of the
second event (rolling a die).
Two events A and B are independent events if the fact that A
occurs does not affect the probability of B occurring.
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15. Addition Rules
Outline
Multiplication Rules
Multiplication Rule 1 (p. 206)
When two events are independent, the probability of both
occurring is
P(A ∩ B) = P(A) · P(B)
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16. Addition Rules
Outline
Multiplication Rules
Remarks on the Multiplication Rule 1
1. Multiplication rule 1 can be extended to three or more
independent events by using the formula
P(A ∩ B ∩ C ∩ · · · ∩ K ) = P(A) · P(B) · P(C ) · · · P(K )
2. In this sequence, the experiments may or may not be the same.
If the experiments are the same, the events may or may not be the
same.
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17. Addition Rules
Outline
Multiplication Rules
Dependent Events (p. 208)
When the outcome or occurrence of the first event A affects the
outcome or occurrence of the second event B in such a way that
the probability is changed, the events A and B are said to be
dependent events.
The conditional probability of an event B in relationship to an
event A is the probability that event B occurs given that the
event A has already occurred. The notation for conditional
probability is P(B|A). This notation does not mean that B is
divided by A; rather, it means the probability that event B occurs
given that event A has already occurred.
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18. Addition Rules
Outline
Multiplication Rules
Multiplication Rule 2 (p. 208)
When two events are dependant, the probability of both occurring
is
P(A ∩ B) = P(A) · P(B|A)
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19. Addition Rules
Outline
Multiplication Rules
Formula for Conditional Probability (p. 210)
The probability that the second event B occurs given that the first
event A has occurred can be found by dividing the probability that
both events occurred by the probability that the first event has
occurred. The formula is
P(A ∩ B)
P(B|A) =
P(A)
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20. Addition Rules
Outline
Multiplication Rules
Conditional Probability and Independent Events
Two events A and B are independent if P(B|A) = P(B) and are
dependent otherwise.
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21. Addition Rules
Outline
Multiplication Rules
Probabilities for “At Least” or “At Most” (P. 213)
In some case, it is easier to compute the probability of the
complement of an event than the probability of the event itself.
This is still true for a sequence of events.
Example: A coin is tossed 5 times. Find the probability of getting
at least one tail. This is equal to 1 minus the probability of
getting no tail at all, which is all heads.
Find the probability of getting at most four tails. This is equal to
1 minus the probability of getting five tails.
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