1. Introduction to Probability
Experiments
Counting Rules
Combinations
Permutations
Assigning Probabilities

2. These are processes that generate well-defined outcomes
Experiments
Experiment
Experimental Outcomes
Toss a coin
Head, tail
Select a part for inspection
Defective, nondefective
Conduct a sales call
Purchase, no purchase
Roll a die
1, 2, 3, 4, 5, 6
Play a football game
Win, lose, tie

3. Probability is a numerical measure of the likelihood of an event occurring
1.0
0.5
Probability:
The occurrence of the event is just as likely as it is unlikely

4. Sample Space For a coin toss: Selecting a part for inspection:
The sample space for an experiment is the set of all experimental outcomes
For a coin toss:
Selecting a part for inspection:
Rolling a die:

5. Counting Experimental Outcomes
To assign probabilities, we must first count experimental outcomes. We have 3 useful counting rules for multiple-step experiments. For example, what is the number of possible outcomes if we roll the die 4 times?
Counting rule for multi-step experiments
Counting rule for combinations
Counting rule for permutations

6. Counting Rule for Multi-Step Experiments
If an experiment can be described as a sequence of k steps with n1 possible outcomes on the fist step, n2 possible outcomes on the second step, then the total number of experimental outcomes is given by:

7. Example: Bradley Investments
Bradley has invested in two stocks, Markley Oil and Collins Mining. Bradley has determined that the possible outcomes of these investments three months from now are as follows.
Investment Gain or Loss
in 3 Months (in $000)
Markley Oil
Collins Mining 10 5
-20 8 -2

8. A Counting Rule for Multiple-Step Experiments
Bradley Investments can be viewed as a
two-step experiment. It involves two stocks, each
with a set of experimental outcomes.
Markley Oil: n1 = 4
Collins Mining: n2 = 2
Total Number of
Experimental Outcomes: n1n2 = (4)(2) = 8

9. Tree Diagram Markley Oil Collins Mining Experimental (Stage 1)
Outcomes
Gain 8
(10, 8) Gain $18,000
(10, -2) Gain $8,000
Lose 2
Gain 10
Gain 8
(5, 8) Gain $13,000
(5, -2) Gain $3,000
Lose 2
Gain 5
Gain 8
(0, 8) Gain $8,000
Even
(0, -2) Lose $2,000
Lose 2
Lose 20
Gain 8
(-20, 8) Lose $12,000
Lose 2
(-20, -2) Lose $22,000

10. Counting Rule for Combinations
This rule allows us to count the number of experimental outcomes when we select n objects from a (usually larger) set of N objects.
Counting Rule for Combinations
The number of N objects taken n at a time is
where
And by definition

11. Example: Quality Control
An inspector randomly selects 2 of 5 parts for inspection. In a group of 5 parts, how many combinations of 2 parts can be selected?
Let the parts de designated A, B, C, D, E. Thus we could select:
AB AC AD AE BC BD BE CD CE and DE

12. Ohio Lottery
Ohio randomly selects 6 integers from a group of 47 to determine the weekly winner. What are your odds of winning if your purchased one ticket?

13. Counting Rule for Permutations
Sometimes the order of selection matters. This rule allows us to count the number of experimental outcomes when n objects are to be selected from a set of N objects and the order of selection matters.

14. Example: Quality Control Again
An inspector randomly selects 2 of 5 parts for inspection. In a group of 5 parts, how many permutations of 2 parts can be selected?
Again let the parts de designated A, B, C, D, E. Thus we could select:
AB BA AC CA AD DA AE EA BC CB BD DB BE EB CD DC CE EC DE and ED

15. Basic Requirements for Assigning Probabilities
Let Ei denote the ith experimental outcome and P(Ei) is its probability of occurring. Then:
The sum of the probabilities for all experimental outcomes must be must equal 1. For n experimental outcomes:

16. Classical Method
This method of assigning probabilities is indicated if each experimental outcome is equally likely

17. Example: Tossing a Die Experimental Outcome P(Ei) 1 1/6 = .1667 2 3 45 6
ΣP(Ei)
1.00

18. Relative Frequency Method
This method is indicated when the data are available to estimate the proportion of the time the experimental outcome will occur if the experiment is repeated a large number of times.
What if experimental outcomes are NOT equally likely. Then the Classical method is out. We must assign probabilities on the basis of experimentation or historical data.

19. Example: Lucas Tool Rental
Relative Frequency Method
Lucas Tool Rental would like to assign probabilities to the number of car polishers it rents each day. Office records show the following frequencies of daily rentals for the last 40 days.
Number of
Polishers Rented
Number
of Days 1 2 3 4 4 6 18 10 2

20. Relative Frequency Method
Each probability assignment is given by dividing the frequency (number of days) by the total frequency (total number of days).
Number of
Polishers Rented
Number
of Days
Probability 1 2 3 4 4 6 18 10 2 40
.10
.15
.45
.25
.05
1.00
4/40

21. Subjective Method When economic conditions and a company’s
circumstances change rapidly it might be
inappropriate to assign probabilities based solely on
historical data.
We can use any data available as well as our
experience and intuition, but ultimately a probability
value should express our degree of belief that the
experimental outcome will occur.
The best probability estimates often are obtained by
combining the estimates from the classical or relative
frequency approach with the subjective estimate.

22. Subjective Method Applying the subjective method, an analyst
made the following probability assignments.
Exper. Outcome
Net Gain or Loss
Probability
(10, 8)
(10, -2)
(5, 8)
(5, -2)
(0, 8)
(0, -2)
(-20, 8)
(-20, -2)
$18,000 Gain
$8,000 Gain
$13,000 Gain
$3,000 Gain
$2,000 Loss
$12,000 Loss
$22,000 Loss
.20
.08
.16
.26
.10
.12
.02
.06