Transcript Document

• Example:
Consider a recent study conducted by the personnel manager of
a major computer software company.
The study showed that 30% of employees who left the firm
within two years did so primarily because they were dissatisfied
with their salary, 20% left because they were dissatisfied with
their work assignments, 12% of the former employees indicated
dissatisfaction with both their salary and their work assignments.
• Question:
What is the probability that an employee who leaves within two
years does so because of dissatisfaction with salary,
dissatisfaction with work assignment or both?
The police force consists of 1200 officers of which 960 are men and
240 are women. Over the past two years, 324 officers on the police
force received promotions.
Table: Promotion status of Police Officers over the past two years.
Promoted
Not Promoted
Total
Men
288
672
960
Women
36
204
240
Total
324
876
1200
Let;
M : Event an officer is a man
W : Event an officer is a woman
A : Event an officer is promoted
Ac : Event an officer is NOT promoted
Q: Find the joint and marginal probabilities.
What is the probability that an officer is promoted given that the
officer is a man?
A committee of female officers raised
a discrimination case on the basis
that 288 male officers had received
promotions but only 36 female
officers had received promotions.
Q: Is this argument true?
If yes; What could be the criteria that
supported this argument?
If not; Why?
Assigning Probabilities
 Basic Requirements for Assigning Probabilities
2. The sum of the probabilities for all experimental
outcomes must equal 1.
P(E1) + P(E2) + . . . + P(En) = 1
where:
n is the number of experimental outcomes
Multiplication Law
The multiplication law provides a way to compute the
probability of the intersection of two events.
The law is written as:
P(A B) = P(B)P(A|B)
Mutual Exclusiveness and Independence
Do not confuse the notion of mutually exclusive
events with that of independent events.
Two events with nonzero probabilities cannot be
both mutually exclusive and independent.
If one mutually exclusive event is known to occur,
the other cannot occur.; thus, the probability of the
other event occurring is reduced to zero (and they
are therefore dependent).
Two events that are not mutually exclusive, might
or might not be independent.
The Sales of Automobiles for 300 days
0
automobile sold 54 days
1
automobile sold 117 days
2 automobile sold 72 days
3 automobile sold 42 days
4 automobile sold 12 days
5 automobile sold 3 days
Total: 300 days
A game is played using one die. If the die is
rolled and shows 1, the player wins $5. If the
die shows any number other than 1, the
player wins nothing. If there is a charge of $1
to play the game, what is the game’s expected
value? What does this value mean?
A construction company is planning to bid on a
building contract. The bid costs the company
$1500. The probability that the bid is accepted is
1/5 . If the bid is accepted, the company will make
$40,000 minus the cost of the bid. Find the
expected value in this situation. Describe what this
value means.
It is estimated that there are 27 deaths
for every 10 million people who use
airplanes. A company that sells flight
insurance provides $100,000 in case of
death in a plane crash. A policy can be
purchased for $1. Calculate the expected
value and thereby determine how much
the insurance company can make over
the long run for each policy that it sells.
.
A 25-year-old can purchase a one-year
life insurance policy for $10,000 at a cost
of $100. Past history indicates that the
probability of a person dying at age 25 is
0.002. Determine the company’s expected
gain per policy.
.
A store specializing in mountain bikes is to
open in one of two malls. If the first mall is
selected, the store anticipates a yearly profit of
$300,000 if successful and a yearly loss of
$100,000 otherwise. The probability of success
is 1/2 . If the second mall is selected, it is
estimated that the yearly profit will be $200,000
if successful; otherwise, the annual loss will be
$60,000. The probability of success at the
second mall is 3/4 . Which mall should be
chosen in order to maximize the expected
profit?
An oil company is considering two sites on which to drill, described as follows:
Site A:
Profit if oil is found: $80 million
Loss if no oil is found: $10 million
Probability of finding oil: 0.2
Site B:
Profit if oil is found: $120 million
Loss if no oil is found: $18 million
Probability of finding oil: 0.1
Which site has the larger expected profit? By how much?
Example-1:
An insurance company sells a 10,000 TRL 1-year term insurance
policy at an annual premium of 290 TRL. Based on many year’s
information, the probability of death during the next year for a
person of customer’s age, sex, health etc. is 0.001
Q: What is the expected gain (amount of money made by the
company) for a policy of this type?
Example -2
Medical Research has shown that a certain type of chemotherapy is successful 70% of
the time when used to treat the skin cancer.
Suppose 5 skin cancer patients are treated with chemotherapy
Let x be the number of successful cures out of five
The probability distribution Table for r.v. X is given as;
X:
0
1
2
3
4
5
F(x):
0.002
0.029
0.132
0.309
0.360
0.168
a.
b.
c.
d.
E(x)=?
St.Dev.(x)=?
Graph of f(x)=? And Interval of µ ± 2σ=?
Use either Chebyshev’s or Empirical to approximate the probability that x falls
in this interval.
e. Compare the results with actual probabilities.
Example: 1
Test the following function to determine whether it is a probability function. If it is
not, try to make it into a probability function
S(x) = (6 - |x – 7|) / 36 , for x = 2, 3, 4, 5, 6, 7, . . . , 11, 12
a. List the distribution of probabilities and sketch a histogram.
b. Do you recognize S(x)? If so, identify it.
Example: 2
The College Board website provides much information for students, parents, and
professionals with respect to the many aspects involved in Advanced Placement (AP)
courses and exams. One particular annual report provides the percent of students who
obtain each of the possible AP grades (1 through 5). The 2008 grade distribution for all
subjects was as follows:
AP Grade Percent
1
20.9
2
21.3
3
24.1
4
19.4
5
14.3
a. ) Express this distribution as a discrete probability distribution.
b. ) Find the mean and standard deviation of the AP exam scores for 2008.
Q:
A clothing manufacturer must decide whether to spend a considerable sum of
money to build a new factory. The following table represent the information
about the profits and deficits called Payoff Table:
New Factory BUILT
New Factory NOT
Built
Good Sales Year
451,000
220,000
Poor Sales Year
- 110,000
22,000
If the clothing manufacturer feels that the probabilities for a good sales year or a
poor sales year are, respectively, 0.40 and 0.60, would building the new factory
maximize his expected profit?
Properties of the Binomial Probability Distributions
1- The experiment consists of a sequence of n identical trials
2- Two outcomes (SUCCESS and FAILURE ) are possible on
each trial
3- The probability of success, denoted by p, does not change
from trial to trial. Consequently, the probability of failure,
denoted by q and equals to 1-p , does not change from trial to
trial
4- The trials are independent.
Example :4
The Heart Association claims that only 10% of adults over 30 can pass the minimum
requirements of Fitness Test. Suppose four adults are randomly selected and each is
given the fitness test.
Use the formula for a binomial random variable to find the probability distribution of
x, where x is the number of adults who pass the fitness test. Graph the distribution.