Probability That Students
Understand
Roxy Peck
Cal Poly, San Luis Obispo
AMATYC 2014
Humor me by doing a quick problem.
Suppose you are one of these 1000 people and you
test positive. Should you be really worried?
If someone tests positive, what is the probability
that they actually have the virus? That is, what
proportion of people who test positive have the
virus?
4
 0.08
Easy, right?
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AMATYC 2014
Here is what this looks like in a typical book
A certain virus infects one in every 200 people.
A test used to detect the virus in a person is
positive 80% of the time if the person has the
virus and 5% of the time if the person does not
have the virus. Let A be the event that “the
person is infected” and B be the event “the
person tests positive.”
Using Bayes’ Theorem, if a person tests
positive, determine the probability that the
person is infected.
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How the student is supposed to solve it…
Let A be the event that “the person is infected”
and B be the event “the person tests positive.”
P(A) = 1/200 = 0.005
P(not A) = 0.995
P(B|A) = 0.80
P(B| not A) = 0.05
Then use
P( A)  P( B | A)
P( A | B) 
P( A)  P( B | A)  P(not A)  P( B | not A)
This is hard for students! But this is the same
problem that people solve intuitively if they have
a data table. We need to connect the two!
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Let’s teach them to do this…
Test Positive Test Negative
Total
Has Virus
4
1
5
DoesNot Have Virus
50
945
995
Total
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946
1000
Same problem, same answer, but easy!
NOT “dumbing” down—students can still do all of
the same types of problems.
Just takes advantage of how people naturally
reason.
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Time to Rethink Probability in the
Intro Stat Course

The good news is that we know more now
about how people understand probability
than we did 15 years ago.

Turns out that people don’t necessarily
find the ideas of probability, like
conditional probability and Bayes rule-like
thinking, hard. What they find hard is
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This stuff
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
The symbols and formulas distract from the
reasoning.

There is now evidence that people have a much
easier time reasoning about natural frequencies
than about relative frequencies, particularly when
the numbers are very small. (Much easier for
people to understand 1 in 10,000 than 0.0001.)

I have become a fan of the hypothetical 1000
approach using tables advocated by Gigerenzer,
Strogatz and others.
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For a great short discussion, see
“Chances Are” article in the New York Times
online by Steve Strogatz
http://opinionator.blogs.nytimes.com/2010/
04/25/chances-are/
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From this article…
“The trick is to think in terms of ‘natural
frequencies’—in simple counts of events
rather than the more abstract notions of
percentages, odds, or probabilities. As
soon as you make this mental shift, the
fog lifts.”
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Interesting study…
From “Calculated Risks” by Gert Gigerenzer.
Told doctors in Germany and the U.S. the
following: a woman in a low risk group
(age 40 to 50 with no family history of
breast cancer) has a positive
mammogram. What is the probability that
she actually has cancer?
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To make the question specific, also
given the following information:
The probability that a woman in this low
risk group has breast cancer is 0.8
percent.
 If a woman has breast cancer, the
probability that she will have a positive
mammogram is 90 percent.
 If a woman does not have breast cancer,
the probability that she will have a
positive mammogram is 7 percent.

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What would the doctors tell the woman with
a positive mammogram?






25 German doctors estimated the probability
that the woman has breast cancer given that
she had a positive mammogram. Estimates
ranged from…
1 percent to 90 percent!!
8 of 25 doctors said 90%.
9 of 25 doctors said between 50 and 80
percent.
8 of 25 said 10 percent or less.
American doctors did worse! About 95% said
somewhere around 75%
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So what is the right answer???
The probability that a woman in this low
risk group has breast cancer is 0.8
percent.
 If a woman has breast cancer, the
probability that she will have a positive
mammogram is 90 percent.
 If a woman does not have breast cancer,
the probability that she will have a
positive mammogram is 7 percent.

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Asked another way…




8 out of 1000 women in this group have breast
cancer.
Of these 8 women with breast cancer, 7 will have
a positive mammogram.
Of the 992 women who do not have breast
cancer, 70 will have a positive mammogram.
Imagine the group of women who would have a
positive mammogram. How many of these
women actually have breast cancer?
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So What Should We Do??
A few questions to think about…
 Do students really need to know the difference between
P(A|B) and P(B|A)?
 Yes! For example, probability that a woman that has
breast cancer tests positive vs. probability that a
woman who tests positive has breast cancer. Can be
very different!
 Do students really need to know the formula for Bayes
Rule?
 Really? Probably not.
 Are we OK with students understanding the difference
between a 1 in 1000 chance and a 1 in 10,000 chance,
even if they don’t fully understand this in terms of
probabilities—0.001 vs. 0.0001?
 I am!
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Classroom Suitable Examples

See Handout for activity and contexts:

Should You Paint the Nursery Pink classroom
activity.

5 contexts for probability problems.
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Should You Paint the Nursery Pink
Some of the questions posed in activity 1:
 How likely is it that a predicted gender is correct?
 Is a predicted gender more likely to be correct when the
baby is male than when the baby is female?
 If the predicted gender is female, should you paint the
nursery pink? If you do, how likely is it that you will need to
repaint?
 Extensions all the way to Bayes Rule problems.
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Five Conditional Probability Contexts
Working with a partner, use one of the
contexts provided to write a question
involving conditional probability that you
could assign in your introductory statistics
class and then use the hypothetical 1000
table approach to solve it.
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Wrap-Up

Do you think that your students would be
able to learn from an activity like the one
discussed in this session (Should you paint
the nursery pink?)?

Do you think that the important ideas of
probability and particularly conditional
probability will be less of a stumbling
block for students than the traditional
formula based approach?
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Thanks!

Thanks for attending this session.

Please feel free to contact me if you have
any questions about the activities or
suggestions for improving them.
rpeck@calpoly.edu
AMATYC 2014