Transcript Chapter 4– Probability Slides 1 to 18 (Text 4.1, 4.2, 4.5)

Chapter 4– Probability
Slides 1 to 18 (Text 4.1, 4.2, 4.5)
Slides 19 – end (Text 4.3, 4.4)
1
Idea of probability

A phenomenon is random if
individual outcomes are
uncertain but there is a
regular distribution of
outcomes in a large number
of repetitions.

Probability of an outcome of
a random phenomenon is
the proportion of time it
would occur in a very long
series of repetitions.
2
Basic definitions

Sample space -- the set of
all possible outcomes of a
random phenomenon.

Event a subset of the
sample space

Probability model


Description of sample
space
A way to assign
probabilities to events
3
Assigning probabilities to
sample spaces

Assign a probability to each
individual outcome. Each
probability is a number
between 0 and 1 and the
probabilities must sum to 1.

The probability of any
event is the sum of the
probabilities of the
outcomes making up the
event.
4
Basic Probability Rules and
Definitions

The probability of any event
is between 0 and 1.

If S is the sample space,
then P(S) =1.

The complement of an
event A is the event, Ac, that
A doesn’t occur.
P(Ac) = 1- P(A)
5
Basic Probability Rules and
Definitions

Two events, A and B,
are disjoint if they
have no elements in
common and, therefore,
can’t occur
simultaneously. We
then have
P(A or B)
= P(A or B occurs) =P(A) + P(B)
6
Another sample
space example

Toss a red die and a black die
once. Define a sample space
and compute some
probabilities.
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Toss a red die and a
black die
Black
1
Red
1 (1,1)
2 (2,1)
3 (3,1)
4 (4,1)
5 (5,1)
6 (6,1)
2
(1,2)
(2,2)
(3,2)
(4,2)
(5,2)
(6,2)
3
(1,3)
(2,3)
(3,3)
(4,3)
(5,3)
(6,3)
4
(1,4)
(2,4)
(3,4)
(4,4)
(5,4)
(6,4)
5
(1,5)
(2,5)
(3,5)
(4,5)
(5,5)
(6,5)
6
(1,6)
(2,6)
(3,6)
(4,6)
(5,6)
(6,6)
8
Addition rule for several
disjoint events

If A, B, and C are disjoint
events then
P(at least one of A, B, C )
= P(A) +P(B) + P(C).

This extends to any number of
disjoint events.
9
Independent
Events




Two events, A and B, are
independent if knowing that
one occurs does not change
the probability that the other
occurs.
If A and B are independent,
then
P(A and B) = P(A)P(B)
This is called the
multiplication rule for
independent events.
This extends to any number of
independent events.
10
Unions

If A and B are events, the
union of A and B, written AB,
is the event that at least one
of A or B occurs. This
definition extends to any
number of events.

Addition Rule
P(A  B) = P(A) +P(B) – P(A and B)

For more events the formula
gets very complex.
11
Example of unions


We have a class of 100
MBA students. 25 take
Accounting (A), 15 take
Finance (F), and 10 take
both. I select a student at
random from the class.
What is the probability
that this student takes at
least one of Accounting or
Finance?
Note – a student might
also take other majors.
12
Conditional
Probability
Conditional Probability
The probability that X occurs
given that Y occurs is written
P(X| Y).
Example – Beer Sales
Montreal Toronto Total
A
150
100 250
B
250
250 500
C
50
200 250
Total
450
550 1000
13

Definition of conditional
probability in terms of
unconditional probabilities

P(X| Y) = P(X and Y)/ P(Y )

Example: 50% of people
entering a store ask for help.
15% of people entering a store
both buy and ask for help. If
someone asks for help, how
likely is it that they will buy?
14
Multiplication
Rule
The multiplication rule is
 P(A and B) = P(B)*P(A| B).

Example: 25% of students
study at least 3 hours per day.
40% of students who study at
least 3 hours per day have a
GPA of 3.0 or better. What is
the probability that a randomly
chosen student studies at least
3 hours per day and has a
GPA of 3.0 or better?
15
An equivalent definition of
independence

Multiplication rule says
P(A and B) = P(B)*P(A| B).

If A and B are independent, then
(1) P(A and B) = P(B)*P(A).

So if A and B are independent, then
(2)
P(A|B) = P(A).

Similarly, if A and B are independent, then
(3)
P(B|A) =P(B)

The bold face conditions (1), (2), (3) are easily
shown to be an equivalent conditions for
independence.

INDEPENDENCE: The likelihood of A occurring
is not influenced by whether or not B occurs
16
Another example –
independence and union rules
used together.
(a)
(b)
Suppose that 50% of
husbands watch a certain
TV program.
Independently of their
husbands, 30% of wives
watch that TV program.
If the husband watches
the program, what is the
probability that the wife
watches it?
What is the probability
that at least one of the
husband or wife watches
the program?
17
Using probability
trees

A ski resort makes a profit 3/4
of the weekends when the
weather is favorable, but only
1/8 of the weekends when the
weather is bad. The
probability is 2/3 that a
weekend has favorable
weather. What is the
probability that the company
makes a profit on a weekend?
18
Baye’s Theorem
Example

In a population of workers,
40% are grade school
graduates, 50% are high
school graduates, and 10%
are college graduates.
10%,5%, and 2% of the
grade school grads, high
school grads, and college
grads, respectively, are
unemployed. A worker is
chosen at random and is
found to be unemployed.
How likely is it that the
worker is a college
graduate?
19
Random variables (Sec 4.3 and
4.4)

A random variable (r.v.) is a
variable whose value is the
numerical outcome of a
random phenomenon.

Discrete r.v. Has a finite
number of possible values

A continuous r.v. can take
any value in an interval of
numbers.
20
21
Roulette


In the game of roulette,
if I bet on dollar on red,
I win a dollar with
probability 18/38.
My total win, W, in 2
plays is a random
variable. Write down its
probability distribution.
22
23
Example: mean and
variance in the roulette
example



Recall – W is my total
win in 2 plays.
Expected value of W =
?
Std. dev of W = ?
24


Continuous random variables (r.v.’s) also have
means, variances and std. deviations. To define
them we need to use calculus ( integrals).
We won’t do that here.
25
The Normal Random Variable
26
Normal Distribution
Example



The time required to
complete a certain
manufacturing job is known
to be normally distributed
with a mean of µ = 20
minutes and a standard
deviation of  = 0.5
minutes.
Then z = (x - µ)/  has the
standard normal
distribution.
Can use this to compute
various probabilities.
27
Uniform
distribution

In the uniform
distribution, any
number between 0 and
1 is equally likely to
occur.
28
Using Excel to generate samples
of random variables

Uniform distribution. The function
= rand()
will generate numbers selected at
random from the uniform distribution.
That is any number between 0 and 1 is
equally likely to be chosen (leave the
space between the brackets blank).

Normal distribution. The function
= norminv(rand(), µ, )
will generate numbers chosen at random
from a normal distribution with mean, µ ,
and standard deviation, .
29
Several random
variables

Often want to look at sums
of several random
variables.
 E.g. Total sales over 10
stores
 E.g. Total return from an
investment “portfolio”
consisting of amounts
invested in 25 different
stocks.

Mean = ?? Std Dev = ???
30

Example:

Two car salesmen. A sells 4 cars
per day on average; B sells 3 cars
per day on average. Each is paid
$30 per day salary, plus a fixed
commission of $75 per car sold.
Expected daily earnings of A= ?
of B =? . Total expected daily
earnings of A and B together = ?

31

Two car salesmen. A
sells 4 cars per day on
average; B sells 3 cars
per day on average.
Each is paid $30 per
day salary, plus a fixed
commission of $75 per
car sold.
32
Investment Example (4.27 on
page 332 of text)


Zadie has invested 20% of
her funds in T-bills, which
have an expected return of
5.2% per year and a std dev
of 2.9%. She has invested
80% of her funds in an
index fund that has an
expected return of 13.3%
per year, but with a std.dev
of 17.0%. The correlation
between T-bills and the
index fund is -0.1.
What is the mean and
standard deviation of the
return on her portfolio?
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