Topic 4.
Sample Space
Fundamental Principle of Counting
Permutations
Combinations
Probability of an Event
Additive Rules
Conditional Probability
Bayes’ Rule
PROBABILITY
Objectives
Count efficiently by applying the
Fundamental Principle of Counting
Count using permutation and combination.
Determine the probability of a given event.
Apply the different laws of probability.
Interpret probability values.
Probability 
a specific term is a
measure of the likelihood that a particular
event will occur.
•
Low Probability 
rare or unusual
occurrences
Event
– a collection of outcome of a
procedure. Outcome we’re looking for.
Simple Event
– a single outcome or one
specific outcome we get from procedure.
Sample Space
– all simple events or
every possible outcome.
Example
Procedure
Simple Event
Sample Space
Flip coin one time
Head
Head or Tail
Flip coin 3 times
1 head, and 2
tails
H, H, H or T,T,T
HHT,
HTH,HTT,TTH,THT,THH
Probability 
the likelihood of an
event occurring, P.
P(A) – probability of event A happening.
3 Type of Probability:
1.Observed Probability

probability
that is estimated based on observation.
“What did happen?”
P(A) = # of times A occurred divided by #
of times procedure was repeated
2.
Classical Probability
– probability based on
the chance of an even occurring. (Each simple
event must have an equal chance of occurring.)
“What should happen”. Comes from “priori”, a
latin word means “before”.
•
P(A) = # of way could occur divided by # of
simple events (outcomes)
3.
Subjective Probability
– an educated guess.
Example:
1.The probability of selecting a heart from a deck of
cards.
P(H) = 13/52 = 0.25
(Classical – what should happen)
2. Flip coin 100 times, you get 64 tails.
P(T) = 64/100 = 0.64 (Observedwhat did happen)
3. Peyton completed 385 out of his first 528 passes.
Find the probability that Peyton will complete a pass.
P(Complete a pass) = 385/528 = 0.729 0r 72.9%
(observed)
•
4.) 260 bolts are examined as they
are produced. Five of them are
found to be defective. On the basis
of this information, estimate the
probability that a bolt will be
defective.
P(B) = 5 / 260 = 0.019.
Knowledge of counting the number of ways
by which events can happen
is important in the study of
probability. The elements of a
sample space can be
systematically listed by
means of a
tree diagram
.
Problem
How many 3digit numbers
can be formed from the digits
1, 2 and 3 if repetition of
digits is not allowed?
Solution:
Hundredth Digit
Tenth Digit
Unit Digit
Number
1
2
3
2
3
3
2
123
132
1
3
3
1
1
2
2
1
213
231
312
321
Answer:
6
numbers
The fundamental principle of
counting provides a rule in
determining the number of
chance occurrence of events.
This is known as the
multiplication rule
.
If one thing can be
done in
n
1
ways and a
second thing can be
done in
n
2
ways, then
the sequence of things
can be done together in
n
1
n
2
ways.
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 Fall '17
 Antonino Magallen