Download Probability

MAT 135
Introductory Statistics and Data Analysis
Adjunct Instructor
Kenneth R. Martin
Lecture 9
October 26, 2016
Agenda
• Housekeeping
– Readings
– HW #5
– Quiz #2
• Chapter 1, 14, 10, 2, 3, & 4
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Housekeeping
•
•
•
•
•
•
Read, Chapter 1.1 – 1.4
Read, Chapter 14.1 – 14.2
Read, Chapter 10.1
Read, Chapter 2
Read, Chapter 3
Read, Chapter 4
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Housekeeping
• HW #5 due.
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Housekeeping
• Quiz #2
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Review
• What have we learned so far ?
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Probability
Definition:
Probability: The likelihood or chance that some
event will take place or happen.
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Probability
Definition:
Outcome: The result of a single trial of a probability
experiment.
Event: A set of outcomes of an experiment.
Sample Space: The set of all possible outcomes of
a probability experiment.
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Probability
Example: Sample Space
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Probability
Example: Sample Space
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Probability
Definition:
Tree Diagram: A visual diagram that creates a
logical step hierarchical process. Helps enumerate
all possible outcomes.
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Probability
Example:
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Probability
Definition: Probability
Probability of any event E
P (E) =
n (E) =
n (S)
# outcomes in E
Total # outcomes in sample space
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Probability
Definition:
Probability is expressed as a decimal.
i.e. Tossing a coin, P(H) = 0.500
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Probability
Theorem 1:
•
Probability of 1.000 means an event is certain to
occur
•
Probability of 0 means the event is certain to NOT
occur.
Therefore:
0  P(E)  1
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Probability
Theorem 2:
If, P(H) = Probability of H occurring
Then
P(not H) = 1.000 - P(H)
or
or
P(H) = 1.000 - P(H)
P(H) = 1.000 - P(H)
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Probability
Theorem 5:
•
The total (sum) of the probabilities, for any discrete
distribution, of all situations equals to 1.000
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Probability
Definition, Theorem 5:
•
Correspondingly, the area under a continuous
probability distribution (normal curve) is equal to
1.000 also.
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Probability
Theorems :
One Event
Out or Two
or More Events
Two or More Events
Out or Two
or More Events
Mutually
Exclusive
Not Mutually
Exclusive
Independent
Dependent
Theorem 3
Theorem 4
Theorem 6
Theorem 7
** An event being a collection of outcomes (i.e. flipping a coin. If an event is (H). 2
outcomes are possible.)
** Mutually exclusive means the occurrence of one event makes the other(s)
impossible
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Probability
Theorem 3:
•
If A and B are mutually exclusive events
•
P (A or B) = P(A) + P(B)
** Called the Additive Law of Probability.
** Whenever there’s an “or”, usually additive.
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Probability
Theorem 3 (example):
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Probability
Theorems :
One Event
Out or Two
or More Events
Two or More Events
Out or Two
or More Events
Mutually
Exclusive
Not Mutually
Exclusive
Independent
Dependent
Theorem 3
Theorem 4
Theorem 6
Theorem 7
** An event being a collection of outcomes (i.e. flipping a coin. If an event is (H). 2
outcomes are possible.)
** Mutually exclusive means the occurrence of one event makes the other(s)
impossible
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Probability
Theorem 4:
•
If A and B are NOT mutually exclusive events (i.e.
they have events in common)
•
P (A or B or Both) = P(A) + P(B) - P(Both)
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Probability
Theorem 4 (example):
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Probability
Theorems :
One Event
Out or Two
or More Events
Two or More Events
Out or Two
or More Events
Mutually
Exclusive
Not Mutually
Exclusive
Independent
Dependent
Theorem 3
Theorem 4
Theorem 6
Theorem 7
** An event being a collection of outcomes (i.e. flipping a coin. If an event is (H). 2
outcomes are possible.)
** Independent event means its occurrence has no influence on the probability of the
other event(s).
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Probability
Theorem 6:
•
If A and B are independent events (i.e. one event
has no influence on the other event(s))
•
P (A and B) = P(A) x P(B)
** Whenever there’s an “and”, it’s multiplication.
** Called the Multiplicative Law of Probability.
*** Check if the scenario has replacement or not.
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Probability
Theorem 6 (example):
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Probability
Theorems :
One Event
Out or Two
or More Events
Two or More Events
Out or Two
or More Events
Mutually
Exclusive
Not Mutually
Exclusive
Independent
Dependent
Theorem 3
Theorem 4
Theorem 6
Theorem 7
** An event being a collection of outcomes (i.e. flipping a coin. If an event is (H). 2
outcomes are possible.)
** Dependent event means its occurrence does have influence on the probability of
the other event(s).
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Probability
Theorem 7:
•
If A and B are dependent events (one event does
has influence on the other event(s))
•
P (A and B) = P(A) x P(B|A)
** P(B|A) The probability of B provided A has occurred.
** Called the Conditional Probability Theorem.
*** Check if the scenario has replacement or not.
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Probability
Theorem 7 (example):
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Probability
Combination Theorems (example):
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Probability
Counting Rules: Fundamental Counting Rule
•
If an event A can happen in any of “a” ways or
outcomes and, after it has occurred, another event
B can happen in “b” ways or outcomes, the
number of ways that both events can happen is:
a*b
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Probability
Counting Rules: Fundamental Counting Rule
(example):
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Probability
Counting Rules: Permutations
•
An ordered arrangement of a set of objects.
n!
P 
(n  r )!
n
r
Pnr = # permutations of n objects taken r of them at a
time
n = total number of objects
r = # objects selected from total
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Probability
Counting Rules: Permutations (example):
n!
P 
(n  r )!
n
r
Q: How many permutations of 4 objects, taken 2 at a time ?
Example: The letters S T O P
n = 4; r = 2
Pnr = (4*3*2*1) / 2 * 1
Pnr = 12
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Probability
Counting Rules: Permutations (example):
ST
SO
SP
TS
TO
TP
OS
OP
OT
PS
PT
PO
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Probability
Counting Rules: Combinations
•
If the way the objects are ordered is not important.
n!
C 
r !(n  r )!
n
r
Cnr = # combinations of n objects taken r at a time
n = total number of objects
r = # objects selected from total
Confidential - Kenneth R. Martin
Probability
Counting Rules: Combinations (example)
n!
C 
r !(n  r )!
n
r
Q: How many combinations of 4 objects, taken 2 at a time ?
Example: The letters S T O P
n = 4; r = 2
Cnr = (4*3*2*1) / 2 * 2
Cnr = 6
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Probability
Counting Rules: Combinations (example):
ST
SO
SP
TO
TP
OP
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