Transcript Understanding Probability and Long-Term

Understanding Probability and Long-Term Expectations
Are Coincidences Improbable?
Example from Student - Kalyani Thampi
I have an example of "chance" that I thought about mentioning in class but we didn't have
enough time.
Recently, I've been receiving letters from collection agencies telling me that I owe X amount of
money on a couple of credit cards at stores where I shop. However, I have never once fallen
behind on payments so I couldn't understand why/how I would be so far in debt with these stores
as to be sent to collection agencies!
After a few weeks of investigation (calling my bank, agency, stores, etc), I solved the mystery:
apparently, there is another Kalyani Thampi who lives on the upper west side on Amsterdam
Avenue with almost the same address, and the stores and collection agencies got it
wrong. Also, she uses the same bank as I do (as I found out later), so when she started paying the
money, it came out of my account!
What are the chances that we have the same name, live in the same neighborhood, shop at the
same two stores and have the same bank?
Kalyani Thampi is a rare name in the U.S. I'd say it's destiny but I wouldn't want to think that it
was my fate to meet a girl with such awful financial literacy.
Understanding Probability and Long-Term Expectations
Randomness and Probability
The probability of any outcome of a random phenomenon is a number between 0 and 1 that
describes the proportion of times that outcome would occur in a very long series of repetitions
Height of US women - Population
Mean height = 65
Standard deviation = 2.5
0.18
0.16
Height of curve
What is the probability that a woman
selected at random from this
population will be less than 61 inches?
Heights of US women
0.14
0.12
0.1
Remaining Area
0.08
Area less 61 inches
0.06
0.04
0.02
0
55
56.3
57.6
58.9
60.2
61.5
62.8
64.1
65.4
66.7
68
69.3
70.6
71.9
73.2
74.5
Answer = 0.05
Understanding Probability and Long-Term Expectations
Probability
What does the word probability mean?
Two distinct interpretations:
• For the probability of winning a lottery based on buying a single ticket -- we can quantify the
chances exactly.
• For the probability that we will eventually buy a home -- we are basing our assessment on
personal beliefs about how life will evolve for us.
The Relative-Frequency Interpretation
Relative-frequency interpretation: applies to situations that can be repeated over and over again.
Examples:
• Buying a weekly lottery ticket and observing whether it is a winner.
•
Observing births and noting if baby is male or female.
Understanding Probability and Long-Term Expectations
Idea of Long-Run Relative Frequency
Probability = proportion of time the event occurs over the long run
Long-run relative frequency of males born in the United States is about 0.512.
Possible results for relative frequency of male births:
Observe the Relative Frequency
Example: observe the relative frequency of male births in a given city over the course of a year.
In 1987 there were a total of 3,809,394 live births in the U.S., of which 1,951,153 where males.
probability of male birth is 1,951,153/3,809,394 = 0.5122
Understanding Probability and Long-Term Expectations
Make an Assumption about the Physical World
Example: assume coins made such that they are equally likely to land with heads or tails up
when flipped - probability of a flipped coin showing heads up is ½.
Understanding Probability and Long-Term Expectations
The Personal-Probability Interpretation
Personal probability: the degree to which a given individual believes the event will happen.
Personal-Probability versus Relative Frequency Probability
Understanding Probability and Long-Term Expectations
Applying Some Simple Probability Rules
Rule 1: If there are only two or more possible outcomes in an uncertain situation, then their
probabilities must add to 1.
Example : If the probability that a particular flight will be on time is 0.70, then it will be late with
probability 0.30.
Rule 2:
If two outcomes cannot happen simultaneously, they are said to be mutually exclusive.
For relative frequency, according to Krantz (1992, p. 190) 25% of all women give birth to their
first child at under 20 years of age.
Age:
Under 20
20 to 24
25 to 29
30 to 34
35 and over
Proportion:
0.25
0.33
0.25
0.125
0.04
Understanding Probability and Long-Term Expectations
Applying Some Simple Probability Rules
Rule 3: If two events do not influence each other, and if knowledge about one doesn’t help with
knowledge of the probability of the other, the events are said to be independent of each other.
If two events are independent, the probability that they both happen is found by multiplying
their individual probabilities.
Example: Suppose for first-year students at your university the percentage is 20%.
If these students are randomly assigned to dorm rooms, what is the probability that two people
assigned to the same room are either both smokers or both nonsmokers?
Independent Events?
Event A is that it snows tomorrow; event B is that the high temperature tomorrow is at least 60
degrees Fahrenheit.
You buy a lottery ticket, betting the same numbers two weeks in a row. Event A is that you win
in the first week; event B is that you win in the second week.
Event A is that a major earthquake will occur somewhere in the world next month; event B is
that the Dow Jones Industrial Average will be higher next month
Understanding Probability and Long-Term Expectations
Applying Some Simple Probability Rules
Rule 4: If the ways in which one event can occur are a subset of those in which another event
can occur, then the probability of the subset event cannot be higher than the probability of the
one for which it is a subset.
Text Question
12. A study by Kahneman and Tversky (1982, p. 496) asked people the following question:
•Linda is 31 years old. single, outspoken, and very bright. She majored in philosophy.
•As a student, she was deeply concerned with issues of discrimination and social justice and also
participated in antinuclear demonstrations.
Please check off the most likely alternative:
A. Linda is a bank teller.
B. Linda is a bank teller and is active in the feminist movement.
Frequency Format versus Probability form
Now let us change the question slightly without changing the description of Linda:
Imagine women who fit the description of Linda
How many of these women are bank tellers?
How many of these women are bank tellers and active in the feminist movement?
Understanding Probability and Long-Term Expectations
When Will It Happen?
•Probability an outcome will occur on any given instance is p.
•Probability the outcome will not occur is (1 – p).
•Outcome each time is independent of outcome all other times.
Probability it doesn’t occur on 1st try but does occur on 2nd try is (1 – p)p.
Understanding Probability and Long-Term Expectations
When Will It Happen?
Example: Number of Births to First Girls
•Probability of a birth resulting in a boy is about .51, and the probability of a birth resulting in a girl
is about .49.
•Suppose couple will continue having children until have a girl.
•Assuming outcomes of births are independent of each other, probabilities of having the first girl
on the first, second, third, fifth, and seventh tries are shown below.