# Reasoning With Probability

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```							 Reasoning With Probability
CMC North 2009
Session 545
Kevin Rees
krees@ma.org
http://courses.ma.org/Math/Rees/index.html
“The amount of data available to help make decisions
in business, politics, research, and everyday life is
staggering: Consumer surveys guide the development
and marketing of product. Polls help determine
political-campaign strategies, and experiments are
used to evaluate the safety and efficacy of new medical
treatments. Statistics are often misused to sway public
opinion on issues or to misrepresent the quality and
effectiveness of commercial products. Students need
to know about data analysis and related aspects of
probability in order to reason statistically – skills
necessary to becoming informed citizens and
intelligent consumers.”
NCTM Principles and Standards for School
Mathematics.
Where do we begin?

NCTM Data Analysis and Probability Standards
for Grades 6 – 8 p 248.
Students should “use proportionality and a
basic understanding of probability to make
and test conjectures about the results of
experiments and simulations.”
Using contingency tables with pre-Algebra or
Algebra I students to investigate relationships
between two categorical variables.

Developing sophistication with percentages
and proportions through comparisons.

First example:
Hospitals and Delayed Discharges
A statistical situation that occurs when two
groups have differing amounts of a key
underlying characteristic, in the previous
example the proportion of major and minor
surgeries performed at each hospital.
A great way to introduce looking for hidden
characteristics (or the idea of confounding) to
students, the idea of a weighted mean, and
for older students concepts of social justice.
One of the best-known examples of Simpson’s
Paradox was the lawsuit against UC Berkeley in
the early 1970s regarding gender bias in their
figures showed a large difference in the
percentage of men accepted to graduate school
compared to women who were accepted. When
admission rates to the individual departments
was analyzed it was found that there was not a
difference in admissions of men and women,
rather that there was a difference in what type of
graduate program each gender tended to apply
to. At the time women applied more frequently
to departments with lower admission rates.
Example 2: Ask Marilyn
STAR Testing
Example 3: STAR Testing and Simpson’s Paradox
A warning …
Simpson’s Paradox does not occur for every
test and all grades between Coleman and the
Dixie schools. What this paradox does remind
us is to be wary of averages or percentages
about two groups when there is another
potentially important characteristic that may
have a profound affect on the results.
“Students should leave secondary school with
the ability to judge the validity of arguments
that are based on data, such as those that
appear in the press.” NCTM Principles and
Standards for School Mathematics.

Mutually Exclusivity and Independence
Example 4: from Innumeracy
Rain on               Rain on
Saturday              Sunday

50%                  50%

Rain on       Rain on
Saturday      Sunday

50%              50%

Rain on
Saturday
Rain on
Sunday
50%
Expected Values
An expected value gives a student a much
better idea of what happens in the long run,
as opposed to a one-off event. Introducing
expected value early in a mathematics
curriculum (pre-Algebra or Algebra I) will allow
students a better understanding of probability
in their high school career.
Expected Values and Testing
Independence
We can use expected values and the
probability of independent events to
make inference about whether two
categorical variables are related or not.

Example 5: Kindergarten and Age
Example 6: Race and the Death Penalty
Statistical Significance: Is the difference observed
between the observed values and the expected values
too large to have occurred naturally (or by chance
alone)?
Expected Values and The Law of Averages
The Oxford American Dictionary, 1980, defines The Law of Averages to be “the proposition
that the occurrence of one extreme will be matched by that of the other extreme so as to
maintain the normal average.”
Here are some common misinterpretations of the Law of Averages:
•Announcer Chick Hearn noted during a Warriors game on December 15, 1990 that Los
Angeles Laker Carl Perkins had made the last six out of six free throws. He then
concluded that “the Law of Averages starts working for Golden State.”
•A roulette wheel has 18 black positions and 18 red positions. A gambler observes five
consecutive reds and then bets heavily on black because “black is due.”
•A family gives birth to their third boy in a row. The mother is distraught since she
wanted to have a girl. The doctor consoles her by telling her that the law of averages
states that it is in her favor to have a boy next time with odds of 100 to 1.
•At a certain university, the head resident of a given dorm states that 25% of incoming
freshman do not graduate within four years. He tells you to look at the four people
nearest to you, and then he states that one of these four will not graduate in four years.

Example 7: The Law of Averages
It is not intuitive to students (and many adults)
that in the long run the proportion of heads
will get closer and closer to 0.5, but the
difference between the number of heads and
the number of tails usually will increase.
100 flips gives 52 heads and 48 tails for a
proportion of heads of .52, and a difference of
4 flips between heads and tails.
10000 flips gives us 50018 heads and 49982 tails
for a proportion of heads of .50018, but a
difference of 36 flips!
Conditional Probabilities and
Expected Values
• Think of a conditional probability as a reduced
sample space – no formulas are needed to
understand.
• Use this idea to approach problems usually
associated with Tree Diagrams and Bayes’ Theorem.
• One more way for students to gain an intuitive
understanding of independence.
Example 8: Newspaper Reading
Example 9: Lie Detector Tests
Students don’t need formulas for conditional
probabilities, such as P(A|B) P(AB), when their
P(B)
intuition will lead them to the same point:
1425/1750 ≈ 81.43% of trustworthy people are


labeled dishonest.
The same goes for proving independence, P(A|B) =
P(A) is less useful than an understanding of what
independence means: the probability a random
person reads the newspaper every day is 358/906 ≈
39.51%, while the probability a male reads the
newspaper every day is 191/358 ≈ 53.35%
Geometric Distribution and Expected Values

Students often struggle with the lottery and the idea of
possible versus probable. Using information about the
California Lottery allows them to investigate the lottery, gain
an understanding of expected values of a random variable,
look at long run behavior with the geometric distribution, and
even incorporate logarithms.

Example 10: Someone must win, why not me?
175711535
Let p  175711536 . So the series can be represented as:

1                       
1 p p2 ... pn1

                         175711536                   

Which breaks down to:

1     1 p n          1 p n
        (1 p)       1 pn
175711536 1 p             1 p
To find when we first hit a probability of 1%, we solve the inequality:

1 p n .01

Manipulating, and then using logarithms, leads us to:


pn .99      nlog p log.99     n log.99 1765958.7
log p
So for 1%, it will take 1,765,959 tickets. For 10% it takes 18,513,045 tickets,
and for 50% it takes 121,793,869 tickets. At this point it would be a better

strategy to buy 50% of the 175,711,536 possible combinations of numbers in
one play.
Combinatorics
• Consider a randomly selected family with two
children. If one child is female, what is the
probability that the other child is male?

• What if the oldest child is female, what is the
probability that the other child is male?
For the first situation:
BB, BG, GB, GG thus the probability is 2/3.

For the second situation:
BB, BG, GB, GG thus the probability is ½.

Example 11: People in the Room
RRR
RRB
RBR
B R R Using this strategy
R B B the group wins 6/8
B R B or 75% of the time
BBR
BBB
The matching problem (commonly seen as the
hat-check problem or the letters in mailboxes
problem)

Example 12: Katie Morag Delivers the Mail
DABC, DACB, DBAC, DBCA, DCAB, DCBA
To solve this for any number of houses is beyond the
typical high school student. In 1751 Euler determined
(without factorial notation!) that
n
P(0 matches in n houses) 1  1  1 ... (1)
!
0! 1 2!           n!
Note that as n gets very large this approaches e-1 or
approximately 0.368 (it is correct to six decimal


places by the time n = 10.) This result is in many
college level introductory probability textbooks.
Resources
Stats: Modeling the World by Bock, Velleman and DeVeaux, 3rd
edition, 2009. Both textbook and teacher resources (which
are excellent.)
Chance News at http://www.dartmouth.edu/~chance/
Innumeracy: Mathematical Illiteracy and its Consequences, by
John Allen Paulos, 1988.
Activity Based Statistics by R. Scheaffer, M. Gnanadesikan, A.
Watkins and J. Witmer, 2004.
Workshop Statistics: Discovery with Data and Fathom by A.
Rossman, B. Chance and R. Lock, 3rd Edition, 2009.
Fifty Challenging Problems in Probability by Frederick Mosteller,
1965.

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