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STAB22 Statistics I
Lecture 14
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Venn Diagrams
Sample Space

Events
Method for representing
events graphically



Sample space (S): rectangle
Outcomes: points in S
Events: areas in rectangle
B
A
Outcomes

Example: Roll of a die

Event A = {5 ,6 } =
= “Roll ≥ 5”
1
2
3
4
5
6
Event A
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Composite Events

Given events A and B, form new events as:
A or B (union, also denoted A U B)


A and B (intersection, also denoted A ∩ B)


Either event A, or event B, or both occur
Both events A & B occur simultaneously
not A (complement, also denoted AC)

Event A does not occur
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Composite Events

Union: A or B


Either event occurs
Intersection: A and B


AUB
Both events occur
A
B
A∩B
Complement: AC

Event A does not occur
A
AC
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Disjoint Events

Events are called disjoint (or mutually
exclusive), if they have no outcomes in
common (i.e. cannot occur simultaneously)
Disjoint (no overlap)
A

B
Not disjoint
A
B
E.g. Which of the events: A, A∩B, AC are always
disjoint?
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Example

Random phenomenon: flipping 2 coins
List outcomes in sample space: S =

List outcomes in each event


A = “1st flip is H” =

B = “2nd flip is T” =

AUB

A∩B

AC
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Probabilities
Probabilities can be assigned to events based on:
 Past experience (empirical probability)


Model (theoretical probability)


E.g. if 5% of microchips produced by a machine
are defective, then P(“microchip defective”) = 0.05
E.g. for fair coin, P(Heads)=1/2
Subjective belief (personal probability)

E.g. I believe my favorite team has 30% probability
of winning Stanley Cup
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Probability Rules
1.
2.
3.
No matter how probabilities are assigned,
they must satisfy the following three rules:
Probability of any event is between 0 and 1:
0 ≤ P(A) ≤ 1, for any event A
Probability of all outcomes is 1
P(S) = 1, where S= sample space
For two disjoint events A & B:
P(A or B) = P(A) + P(B)
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Example

Roll fair 6-sided die


Sample space S = {1, 2, 3, 4, 5, 6}
Each outcome has equal probability 1/6

Find probability of A=“roll 1 or 2”

Find probability of B=“roll even number”
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Complement Rule

Probability of AC can be found from that of A
using complement rule:
PA

C
  1 P  A
A
AC
Why?



Are events A and AC mutually exclusive?
What is the union A or AC
What is the probability of A or AC
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General Addition Rule

For any events A, B
A
B
P  A or B  
 P  A   P  B   P  A and B 

If events A, B are disjoint
A
B
P  A and B   0 (no overlap) 
 P  A or B   P  A   P  B 
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Example

Find probability that a card chosen at
random from a standard deck of cards will
be either a king or a heart?
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Example

Shop accepts either AmEx or VISA credit
cards. A total of 24% of its customers carry
AmEx, 61% carry VISA, and 11% carry both.
What percentage of its customers, carry a
card that the shop accepts?
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Example

Weather forecaster predicts following
probabilities for tomorrow’s weather:


P(sun) = .6, P(temp < 0°C) = .4, and
P(sun and temp < 0°C) = .5
Are these valid probs?
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Conditional Probability

Conditional Probability: probability of event
A, given that event B has occurred


Denoted by P(A | B) (vertical bar read as “given”)
Probability of A, if all of the possible outcomes are
restricted only to the ones in B
1
3
5
4
2
Condition
on event B
2
4
6
6
reduced
sample
space B
A∩B
B
A
S
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Conditional Probability

Conditional probability given by formula
P  A and B 
P  A | B 
P B

Example: Roll of a die



Let A = {3, 4} and B = {2, 4, 6}
P(A | B) is prob of {3, 4} given roll is even
Find P(A | B)
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Example

Consider contingency table
of law school applicants:






Male
Female
total
Accept Reject total
20
25
5
30
75
45
35
65 100
Define events: M=male, F=female, A=accept, &
R=reject; and find the following (empirical) probs
P(M) =
● P(F) =
● P(A) =
● P(R) =
P(A and M) =
● P(A and F) =
P(A | M) =
P(A | F) =
P(M | A) =
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