Main rules for working with
probabilities:
Chapter 7
1.  P(Ac) = 1-P(A)
2.  (a). P(A or B) = P(A) + P(B) – P(A and B)
(b). P(A or B) = P(A) + P(B)
3. (a). P(A and B) = P(A)P(B|A)
(b). P(A and B) = P(A)P(B)
(c). P(A and B and C) = P(A)P(B)P(C)…
4. P(B|A) = P(B and A)/P(A)
Probability
part 2
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
7.5
2
Strategies for Finding
Complicated Probabilities
Hints and Advice
for Finding Probabilities
Example 7.2 Winning the Lottery
•  P(A and B): define event in physical terms and see if
know probability. Else try multiplication rule (Rule 3).
Event A = winning number is 956. What is P(A)?
Method 1: With physical assumption that all 1000
possibilities are equally likely, P(A) = 1/1000.
•  Series of independent events all happen: multiply
all individual probabilities (Extension of Rule 3b)
Method 2: Define three events,
B1 = 1st digit is 9, B2 = 2nd digit is 5, B3 = 3rd digit is 6
Event A occurs if and only if all 3 of these events occur.
Note: P(B1) = P(B2) = P(B3) = 1/10. Since these events
are all independent, we have P(A) = (1/10)3 = 1/1000.
•  One of a collection of mutually exclusive events
happens: add all individual probabilities (Rule 2b
extended).
•  Check if probability of complement easier,
then subtract it from 1 (applying Rule 1).
* Can be more than one way to find a probability.
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
3
4
Steps for Finding Probabilities (my version:)
Hints and Advice for Finding Probabilities
Step 1: (a). List each separate random circumstance
involved in the problem.
•  None of a collection of mutually exclusive events
happens: find probability one happens, then subtract
that from 1.
(b). List the possible outcomes for each random
circumstance.
•  Conditional probability: define event in physical
terms and see if know probability. Else try Rule 4 or
next bullet as well.
(c). Assign whatever probabilities you can with the
knowledge you have.
Step 2: Write the event for which you want to determine
the probability in terms of the outcomes in Step 1. Pay
attention to words like not, and, or, given.
•  Know P(B|A) but want P(A|B): use tree diagrams!
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
5
Step 3: Based on the key words in step 2 determine which
probability rules can be combined to find the probability
of interest. Look for any combination of ‘independent’,
‘disjoint’, ‘conditional probability’…
6
1
Example: A particular brand of cereal box
contains a prize in each box. There are
four possible prizes, and any box is equally
likely to contain each of the four prizes.
What is the probability that you will
receive two different prizes if you purchase
two boxes?
Example: Assume that the probability that
each birth in a family is a boy is 0.512 and
that the outcomes of successive births are
independent. If the family has three
children, what is the probability that they
have two boys and one girl (in any order)?
7
Tree Diagrams
Step 1: Determine first random circumstance in sequence, and
create first set of branches for possible outcomes. Create one
branch for each outcome, write probability on branch.
Step 2: Determine next random circumstance and append branches
for possible outcomes to each branch in step 1.
Write associated conditional probabilities on branches.
Step 3: Continue this process for as many steps as necessary.
Step 4: To determine the probability of following any particular
sequence of branches, multiply the probabilities on those
branches. This is an application of Rule 3a.
Step 5: To determine the probability of any collection of sequences
of branches, add the individual probabilities for those sequences,
as found in step 4. This is an application of Rule 2b.
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
9
Example (con’t): the hypothetical two-way
table
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
8
Example: In a 1998 survey of most of the 9th grade
students in Minnesota, 22.9% of boys and 4.5% of
girls admitted that they gambled at least once a
week during the previous year. The population
consisted of 50.9% girls and 49.1% boys.
(a). What is the probability that a randomly selected
student gambles weekly?
(a). What is the probability that a randomly selected
student will be a male who also gambles at least
weekly?
(b). Suppose that you select a student and find that the
student is a gambler. What is the probability that
10
the student is a girl?
Example 7.8 Teens and Gambling (cont)
9th
Sample of
grade teens: 49.1% boys, 50.9% girls.
Results: 22.9% of boys and 4.5% of girls admitted
they gambled at least once a week during previous year.
Start with hypothetical 100,000 teens …
(.491)(100,000) = 49,100 boys and thus 50,900 girls
Of the 49,100 boys, (.229)(49,100) = 11,244
would be weekly gamblers.
Of the 50,900 girls, (.045)(50,900) = 2,291
would be weekly gamblers.
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
P(boy and gambler) =
P(girl | gambler) =
P(gambler) =
11
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
12
2
Example:
Last week, Alicia went to her physician for a routine medical
exam. This morning her physician phoned to tell her that one
of her tests came back positive, indicating that she may have a
disease D.
The physician told her that the the test is 95% accurate
whether someone has disease D or not. In other words, when
someone has disease D, the test detects it 95% of the time.
When someone does not have D, the test is rightly negative
95% of the time.
Therefore, according to the physician, even though only 1 in
1000 women of Alicia’s age actually has D, the test is a pretty
good indicator that Alicia actually has disease D.
Given the positive test result, what is the probability that Alicia
has disease D?
13
Medical testing:
•  The sensitivity of a test is the proportion of
people who correctly test positive when
they actually have the disease.
•  The specificity of a test is the proportion of
people who correctly test negative when
they don’t have a disease.
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
7.6
Example (con’t):
14
Using Simulation to
Estimate Probabilities
Some probabilities so difficult or timeconsuming to calculate – easier to simulate.
ROUTINE MEDICAL TESTING
VS.
POPULATION AT RISK
KEY QUESTIONS: What is the baseline risk in the population?
What is the population?
15
Example 7.19 Getting All the Prizes
If you simulate the random circumstance n times
and the outcome of interest occurs in x out of
those n times, then the estimated probability
for the outcome of interest is x/n.
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
16
7.7
Coincidences & Intuitive
Judgments about Probability
Cereal boxes each contain one of four prizes.
Any box is equally likely to contain each of the four prizes.
If buy 6 boxes, what is the probability you get all 4 prizes?
1.  Confusion of the Inverse
2.  Specific People vs. Random Individuals
3.  Coincidences
Shown above are 50 simulations of generating a set of 6 digits,
each equally likely to be 1, 2, 3, or 4. There are 19 bold
outcomes in which all 4 prizes were collected. The estimated
probability is 19/50 = .38. (Actual probability is .3809.)
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
17
4.  The Gambler’s Fallacy
18
3
Confusion of the Inverse
Example: Diagnostic Testing
Confuse the conditional probability “have the disease”
given “a positive test result” -- P(Disease | Positive),
with the conditional probability of “a positive test result”
given “have the disease” -- P(Positive | Disease), also
known as the sensitivity of the test.
Often forget to incorporate the base rate for a disease.
19
Most of the physicians to whom this question
was posed answered that the probability
was truly malignant was about 75%. What
is the actual probability?
What is the baseline risk in the population in
this example?
A study of 100 physicians:
One of your patients has a lump in her breast. You
are almost certain that it is benign, in fact you
would say there is only 1% chance that it is
malignant. But just to be sure, you have the
patient undergo a mammogram.
You know from the medical literature that
mammograms are 80% accurate for malignant
lumps and 90% accurate for benign lumps.
Sadly, the mammogram for your patient is returned
with the news that the lump is malignant. What
are the chances that it is truly malignant?
20
Specific People versus
Random Individuals
The chance that your marriage will end in divorce is 50%.
Does this statement apply to you personally?
If you have had a terrific marriage for 30 years, your
probability of ending in divorce is surely less than 50%.
Two correct ways to express the aggregate divorce statistics:
•  In long run, about 50% of marriages end in divorce.
•  At the beginning of a randomly selected marriage,
the probability it will end in divorce is about .50.
21
Coincidences
22
The Gambler’s Fallacy
A coincidence is a surprising concurrence of events,
perceived as meaningfully related, with no apparent
causal connection.
The gambler’s fallacy is the misperception of
applying a long-run frequency in the short-run.
Example 7.23 Winning the Lottery Twice
In 1986, Ms. Adams won the NJ lottery twice in a short time
period. NYT claimed odds of one person winning the top prize
twice were about 1 in 17 trillion. Then in 1988, Mr. Humphries
won the PA lottery twice.
•  Primarily applies to independent events.
•  Independent chance events have no memory.
Example:
1 in 17 trillion = probability that a specific individual who
plays the lottery exactly twice will win both times.
Making ten bad gambles in a row doesn’t change
the probability that the next gamble will also be bad.
Millions of people play the lottery. It is not surprising
that someone, somewhere, someday would win twice.
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
23
Copyright ©2006 Brooks/Cole, a division of Thomson Learning, Inc.
24
4