Download Lecture 22

Probability: First Steps, and
Human Intuitions!
CSCI 2824, Fall 2012!
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Assignments
• For this week: Read Chapter 6, sections 1-3
• Problem Set 4 due today!
• Problem Set 5 (which will be a short one) will be sent
out next week and due the final day of class.
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Initial Ideas
•  Imagine we have a sample space of possible
events. The sample space should exhaust all
the possible outcomes of events.
•  For example, if you roll a die, the sample
space would be the set of possible
outcomes:
{ 1, 2, 3, 4, 5, 6}
•  The most natural kind of sample space is
one in which we assume all outcomes are
equally likely (like our die roll above).
•  The probability of a particular event in this
case can be defined as the number of
outcomes corresponding to this event
divided by the number of possible
outcomes.
Quickie Examples
•  Probability of getting a 2 on a die roll:
[Number of ways of rolling a 2]/6
= 1/6
•  Probability of getting an even number on a
die roll:
Size{2, 4, 6}/6
= |{2, 4, 6}|/6
= ½
•  Probabilities are thus always in the range:
[0, 1]
•  The probability of the union (“or”) of a bunch
of distinct events is simply the sum of their
individual probabilities
•  The probability of the intersection (“and”) of
a bunch of distinct events is the product of
their individual probabilities. (Note: by
“distinct”, we mean that the events have no
effect on each other. In probability, these are
called independent events.)
Rolling two dice
Probability of rolling a 2:
|{(1,1)}| / 36 = 1/36
Probability of rolling a 5:
| {(1, 4), (2, 3), (3, 2), (4, 1)}|/ 36 = 1/9
Probability of rolling a 2 or a 5: 1/36 + 1/9 = 5/36
Now that we know some
combinatorics, we can do some
interesting probability problems:
• What’s the probability of being dealt a flush
(all cards of the same suit) in poker?
• Assuming all Mastermind codes are equally
probable, what is the probability of seeing a
code without an orange peg?
• What’s the probability of rolling a full house
on the first roll in Yahtzee?
“The Birthday Problem”
•  In a room of N people, you ask everyone for
their birthday, and you make a bet that at
least two people will have an identical
birthday. How big does N have to be before
the probability is greater than 0.5 for at least
two people to have the same birthday? (That
is, how many people need to be in the room
before you can expect to win your bet?)
A Yahtzee-like problem: step 1
•  I roll 3 dice. •  What is the probability of all three coming
up with the same number?
•  What is the probability of just two coming
up with the same number (and the third
something else)?
•  What is the probability of three distinct
numbers?
Let’s try a Yahtzee-like problem,
but with only three dice
•  I want to roll three of a kind, and I have two rolls
to do it.
•  If I roll three of a kind on the first roll, I quit.
•  If I roll two of a kind on the first roll (say, 2 3’s
and a 5), then I’ll roll the non-matching die for my
second roll.
•  If I roll all three different numbers, I’ll just roll
again.
•  What’s my chance of winning?
The Let's Make A Deal Problem
• Behind one door is a great prize. (The usual "great
prize" on the old Let's Make a Deal show was a car.)
• The other two doors conceal a "Zonk" prize. (The
usual "Zonk prize" on the old show was a goat or
some other farm animal.)
• You get to choose one door.
• Once you have chosen, Monty Hall will open a
"Zonk" door among the remaining two doors, and
will ask "Do you want to change your choice to
another door?"
Should you change your choice?
Example: You choose Door No. 1. Monty opens up
Door 2 (revealing the inevitable goat) and asks if you
want to switch to Door No. 3.
Let's Make A Deal (The Manic Version)
• You choose a door among these 24.
• Monty throws open 22 Zonk doors and asks "Do
you want to switch?"
Here’s a question that touches on
human thinking about probability…
•  I’m playing a game with three dice. In this
case, my goal is just to roll the largest
possible number.
•  A couple of warm-up questions: what’s the
probability of rolling a 4? What’s the
probability of rolling a 17?
Now, imagine the following
scenario.
•  Mike rolls a 4 with the three dice. •  What’s the probability that Mike will do
better on the next roll?
Now, imagine the following
scenario.
•  Mike rolls a 17. •  What’s the probability that Mike will do
worse on the next roll?
Now, imagine the following
scenario.
•  Mike rolls a 4 with the three dice.
•  Mike yells at the dice, “You stupid dice!” •  What’s the probability that Mike will do
better on the next roll?
Now, imagine the following
scenario.
•  Mike rolls a 17.
•  Mike praises the dice: “You wonderful
dice!” •  What’s the probability that Mike will do
worse on the next roll?
What’s the upshot of all this?
•  It might be easy to think that “yelling helps
to improve the dice” and that “praise does
no good in improving the dice”
•  This is a simple instance of what in statistics
is referred to as “regression to the mean”.
Regression to the mean: the “Sports Illustrated” effect
The Conjunction Effect!
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!Bill is 34 years old. He is intelligent, but
unimaginative, compulsive, and
generally lifeless. In school, he was
strong in mathematics but weak in
social studies and humanities.!
Bill is a doctor, and his hobby is playing poker.!
Bill is an architect.!
Bill is an accountant.!
Bill plays jazz for a hobby.!
Bill surfs for a hobby.!
Bill is a reporter.!
Bill is an accountant who plays jazz for a hobby.!
Bill climbs mountains for a hobby.!
Typicality Effects
Which sequence of coin tosses is more likely?
HHHHHHHH
HTTHTHTH
Memory Effects
Estimate the proportion of English words that begin
with the letter "K" versus words that have a "K" in the
third position.
A Famous Experiment on
“Guessing” by Gerd Gigerenzer
We create a test on American cities
(populations) with lots of questions of the
form:
“Which is bigger: SAN JOSE or SAN
ANTONIO?”
We then administer this test to a classroom of
American students and a classroom of German
students; the German students do better.
A Famous Experiment on
“Guessing” by Gerd Gigerenzer
Now, we create a test on German cities
(populations) with lots of questions of the
form:
“Which is bigger: DORTMUND or BREMEN?”
We then administer this test to a classroom of
American students and a classroom of German
students; now the American students do better.
"
"
Suppose we have to choose between pairs drawn from a
list of 100. Further suppose:"
"
a. When both objects are recognized, we have a 60
percent chance of getting the right answer. (E.g., is
Munich a bigger city than Berlin?)"
"
b. When both objects are unrecognized, we have a 50
percent chance. (Essentially, we’re just “flipping a coin”:
is Dortmund bigger than Duisberg?)"
"
c. When one object is unrecognized, we have an 80
percent chance of getting the right answer. (Is Munich
bigger than Dortmund?)"
"
Three people take the test, which has 100 * 99 /2 = 4950 questions.
One (person A) recognizes each and every object in the list. His
score is:!
.6 * (100 * 99 / 2) = 2970!
Person B doesn’t know a thing about the objects in the list. His
score is:!
0.5 * (100 * 99/2) = 2475!
Person C knows half the list. His score is:!
0.5 * (50 * 49 / 2) + 0.6 * (50 * 49 / 2) + 0.8 * (50 * 50) = !
612.5 + 735 + 2000 = 3347.5!
Moral: A little ignorance can sometimes help.!