Chapter 2
Probability
Motivation
We need concept of probability to make
judgments about our hypotheses in the
scientific method. Is the data consistent with
our hypotheses?
 Example:
Suppose an old drug cures 50 percent of the
time. Our new drug cures 6 out of 10 patients.
Is our new drug better ?

Probabilities…



The probability of any outcome of a random
phenomenon is the proportion of times the outcome
would occur in a very long series of repetitions.
Probability is a long-term frequency.
List the outcomes of a random experiment…
Randomness


…a random experiment is an action or process
that leads to one of several possible outcomes.
Example 1:
Experiment
Outcomes
Flip a coin
Heads, Tails
Exam Marks
Numbers: 0, 1, 2, ...,
100
Assembly Time
t > 0 seconds
Course Grades
F, D, C, B, A, A+
Ex2 Probability Trees : Flip a coin
Heads
P(HH) = 1/4
.5
.5
Heads
P(HT) = 1/4
Tails
.5
.5
P(TH) = 1/4
Heads
Tails
.5
.5
Tails
P(TT) = 1/4
Ex3. Two Coin Flips



P(No Heads) = P(TT) = ¼
P(One Head) = P(HT) + P(TH) = 1/4 +1/4=
1/2
P(Two Heads) = P(HH) = 1/4
Example


This list must be exhaustive, i.e. ALL possible
outcomes included.
Ex : Die roll {1,2,3,4,5}
Die roll {1,2,3,4,5,6}
The list must be mutually exclusive, i.e. no two
outcomes can occur at the same time:
Ex : Die roll {odd number or even number}
Die roll{ number less than 4 or even number}
Sample Space & Event




Set of all possible outcomes is called the sample space, S.
Ex : Coin toss: S={H, T}, Die toss: S={1, 2, 3, 4, 5, 6}
Toss coin twice: S={HH, TT, HT, TH}
An individual outcome of a sample space is called a
simple event
An event is a collection or set of one or more simple
events in a sample space.
Example :
Roll of a die: S = {1, 2, 3, 4, 5, 6}
Simple event: the number “3” will be rolled
Event: an even number (one of 2, 4, or 6) will be rolled
Properties of Probabilities…
(1)
(2)
The probability of any outcome is between 0
and 1
0 ≤ P(Oi) ≤ 1 for each i, and
The sum of the probabilities of all the
outcomes equals 1
P(O1) + P(O2) + … + P(Ok) = 1
P(Oi) represents the probability of outcome i
Events & Probabilities…


The probability of an event is the sum of the
probabilities of the simple events that constitute
the event.
Ex :
(assuming a fair die) S = {1, 2, 3, 4, 5, 6} and
P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6
Then:
P(EVEN)
= P(2) + P(4) + P(6) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2
Equally Likely Outcomes



Probabilities under equally likely outcome case is
simply the number of outcomes making up the
event, divided by the number of outcomes in S.
Example:
A die toss, A={2, 4, 6}, so
P(A) = 3/ 6 = 1/2 = .5
Example:
Coin toss, A={H}, P(A) = 1/2= .5
Classical Approach…


If an experiment has n possible outcomes, this
method would assign a probability of 1/n to
each outcome.
Example :
Experiment: Rolling a die
Sample Space: S = {1, 2, 3, 4, 5, 6}
Probabilities: Each sample point has a 1/6
chance of occurring.
6.12

Experiment: Rolling dice
Sample Space: S = {2, 3, …, 12}
Probability Examples:
P(2) = 1/36
P(7) = 6/36
P(10) = 3/36
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12

Bits & Bytes Computer Shop tracks the number
of desktop computer systems it sells over a
month (30 days):
For example,
10 days out of 30
2 desktops were sold.
From this we can construct
the probabilities of an event

Desktops Sold
# of Days
0
1
1
2
2
10
3
12
4
5
Relative Frequency Approach…
Desktops Sold
# of Days
Desktops Sold
0
1
1/30 = .03
1
2
2/30 = .07
2
10
10/30 = .33
3
12
12/30 = .40
4
5
5/30 = .17
∑ = 1.00
“There is a 40% chance Bits & Bytes will sell 3
desktops on any given day”
Conditional Probability…


Conditional probability is used to determine
how two events are related; that is, we can
determine the probability of one event given
the occurrence of another related event.
Conditional probabilities are written as P(A |
B) and read as “the probability of A given B” and
is calculated as:

Again, the probability of an event given that
another event has occurred is called a
conditional probability…
Note how “A given B” and “B given A” are related…
Conditional Probability

Example:
Die toss, A={1, 2, 3}, B={2, 4, 6}.
P(A | B) = P(A and B) / P(B) = (1/6) / (3/6) = 1/3.
Example
Why are some mutual fund managers more successful
than others? One possible factor is where the manager
earned his or her MBA. The following table compares
mutual fund performance against the ranking of the
school where the fund manager earned their MBA:
Mutual fund outperforms
the market
Mutual fund doesn’t
outperform the market
Top 20 MBA program
.11
.29
Not top 20 MBA program
.06
.54
E.g. This is the probability that a mutual fund
outperforms AND the manager was in a top20 MBA program; it’s a joint probability.
Alternatively, we could introduce shorthand
notation to represent the events:




A1 = Fund manager graduated from a top-20 MBA program
A2 = Fund manager did not graduate from a top-20 MBA
program
B1 = Fund outperforms the market
B2 = Fund does not outperform the market
A1
A2
B1
B2
.11
.29
.06
.54
Ex : P(A2 and B1) = .06
= the probability a fund outperforms the market
and the manager isn’t from a top-20 school.

Example : What’s the probability that a fund will
outperform the market given that the manager
graduated from a top-20 MBA program?
Recall:
A1 = Fund manager graduated from a top-20 MBA
program
A2 = Fund manager did not graduate from a top-20
MBA program
B1 = Fund outperforms the market
B2 = Fund does not outperform the market

Thus, we want to know “what is P(B1 | A1) ?”

We want to calculate P(B1 | A1)
B1
B2
P(Ai)
A1
A2
.11
.29
.40
.06
.54
.60
P(Bj)
.17
.83
1.00
Thus, there is a 27.5% chance that that a fund will outperform
the market given that the manager graduated from a top-20
MBA program.
Marginal Probabilities…

Marginal probabilities are computed by adding across
rows and down columns; that is they are calculated in
the margins of the table:
P(A2) = .06 + .54
“what’s the probability a fund
manager isn’t from a top school?”
B1
B2
P(Ai)
A1
A2
.11
.29
.40
.06
.54
.60
P(Bj)
.17
.83
1.00
P(B1) = .11 + .06
“what’s the probability a fund
outperforms the market?”
BOTH margins must add to 1
(useful error check)
Probability Problems


P(Married)
= 59,920/103,870
P(Married | 18-29)
= 7842/ 22,512
Independence…

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One of the objectives of calculating conditional
probability is to determine whether two events are
related.
In particular, we would like to know whether they are
independent, that is, if the probability of one event is
not affected by the occurrence of the other event.
Two events A and B are said to be independent if
P(A|B) = P(A)
or
P(B|A) = P(B)
Independence


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Events A and B are independent if:
P(A|B) = P(A), that is, if the additional
knowledge of B does not change the probability
of A happening.
On the marriage problem, the events Marriage
and 18-29 are dependent because
P(Marriage|18-29) does not equal P(Marriage),
so these events are dependent.
In die tossing, event roll a 3 on roll 2 is
independent of roll a 6 on roll 1.
P(3 | 6) =1/6 =P(3) = 1/6.
Independence…
For example, we saw that
P(B1 | A1) = .275
The marginal probability for B1 is: P(B1) = 0.17
 Since P(B1|A1) ≠ P(B1), B1 and A1 are not
independent events.
 Stated another way, they are dependent. That
is, the probability of one event (B1) is affected
by the occurrence of the other event (A1).
