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Basic Probability
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Understanding Probability: Concepts, Calculations, and Bayes' Theorem, Study notes of Mathematics
An introduction to probability theory, covering topics such as sample spaces, events, conditional probability, and bayes' theorem. It includes explanations of concepts, formulas for calculating probabilities, and examples using contingency tables and tree diagrams.
Typology: Study notes
Pre 2010
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Basic Probability
Topics
- Basic probability concepts
- Sample spaces and events, simple
probability, joint probability
- Conditional probability
- Statistical independence, marginal
probability
- Bayes’s theorem
Events
- Simple event
- Outcome from a sample space with one
characteristic
- e.g.: A red card from a deck of cards
- Joint event
- Involves two outcomes simultaneously – e.g.: An ace that is also red from a deck
- Joint event
of cards
Visualizing Events
- Contingency tables • Tree diagrams
Red
2
24
26
Black
2
24
26
Total
4
48
52
Ace
Not Ace
Total
FullDeckof Cards
RedCards BlackCards
Not an AceAce Ace
Not an Ace
The event of a triangle
AND
blue in color
Joint Events
Two triangles that are blue
Special Events
• Impossible event
e.g.: Club & diamond on one card
draw
• Complement of event
- For event A, all events not in A– Denoted as A’– e.g.: A: queen of diamonds
A’: all cards in a deck that are not queen of diamonds
Null Event♣
Contingency Table
A Deck of 52 Cards
Ace
Not an
Ace
Total
RedBlackTotal
2
24
2
24
26 26
4
48
52
Sample Space
Red Ace
FullDeckof Cards
Tree Diagram
Event Possibilities
RedCards BlackCards
Ace Not an AceAce Not an Ace
(There are 2 ways to get one 6 and the other 4)
e.g. P
= 2/
- The probability of an event E:• Each of the outcomes in the sample space
is equally likely to occur
Computing Probabilities
number of event outcomes
(^
)^
total number of possible outcomes in the sample space
P E
X T
Computing Joint Probability
- The probability of a joint event, A and B:
(^
and
) =
(^
)
number of outcomes from both A and B
total number of possible outcomes in sam
p
le space
P
A
B
P A
B ∩
=
E.g.
(Red Card and Ace)
2 Red Aces
1
52 Total Number of Cards
26
P^ =
=
Computing Compound
Probability
- Probability of a compound event, A or B:
(^
or
)^
(^
)
number of outcomes from either A or B or both
total number of outcomes in sample space
P A
B
P A
B
=
∪
= E.g.
(Red Card or Ace)4 Aces + 26 Red Cards - 2 Red Aces
52 total number of cards
28
7
52
13
P = =
=
P(A
) 1
P(B
) 2
P(A
1
and B
)
Compound Probability
(Addition Rule)
P(A
1
or B
1
) = P(A
) + P(B 1
) - P(A 1
1
and B
)^1
P(A
1
and B
)
Total
Event
P(A
2
and B
) 1
Event
Total
A
1 A
2
B
1
B
2
P(B
) 1
P(A
2
and B
) P(A
) 2
For Mutually Exclusive Events: P(A or B) = P(A) + P(B)
Conditional Probability Using
Contingency Table
Black
Color
Type
Red
Total
Ace
2
2
4
Non-Ace
24
24
48
Total
26
26
52
Revised Sample Space
(Ace and Red)
2 / 52
2
(Ace | Red)
(Red)
26 / 52
26
P
P
P
=
=
=
Conditional Probability and
Statistical Independence
- Conditional probability:• Multiplication rule:
(
and
)
(^
|^
)
(
)
P A
B
P A
B
P B
=
(
and
)
(
|^
)
(
)
(
|^
)
(
)
P
A
B
P A B
P B
P
B
A
P A
= =