Transcript PowerPoint - Penn State University

STATISTICS 200
Lecture #12
Thursday, September 29, 2016
Textbook: Sections 7.3, 7.4, 7.7
Objectives:
• Identify complementary events and handle probability calculations
• Identify mutually exclusive events and handle probability calculations
• Identify independent events and handle probability calculations
• Understand that conditioning on a particular individual can change risks
• Develop a sense for why seeming coincidences occur frequently
• Identify, and resist the temptation to fall for, the “gambler’s fallacy”
• Understand a common situation where confusion of the inverse occurs
This week…
Randomness
Interpretations of probability
7 Probability
• Relative Frequency
• Personal Probability
Sample spaces and events
Thursday
Flawed intuition
More probability
practice
• Complementary
• Mutually Exclusive
• Dependent / independent
Basic Rules
• Complement rule
• Addition rule
• Multiplication rule
Summary of Rules
Rule 1: Complement Rule
P(A) + P(Ac) = 1 if Ac represents the complement of A
Rule 2B: Additive Rule
P(A or B) = P(A) + P(B) if Events A and B are mutually exclusive
Note: two events that are complements are always mutually
exclusive
Rule 3B: Multiplication Rule
P(A and B) = P(A)×P(B) if Events A and B are independent
Example
Maria wants to take French or Spanish, or both. But classes
are closed, ands he must apply to enroll in a language
class. She has a 60% chance of being admitted to French,
a 50% chance of being admitted to Spanish, and a 20%
chance of being admitted to both French and Spanish. If
she applies to both French and Spanish, the probability that
she will be enrolled in either French or Spanish (or possibly
both) is….
French
0.6
P(French) = ______
0.5
P(Spanish) = ________
0.2
P(French and Spanish) = ______
Spanish
Example
The probability that she will be enrolled in either French or
Spanish (or possibly both) is….
P(French) +P(Spanish) – P(both)
P(French or Spanish) = __________
0.6 + 0.5 – 0.2
= _______
0.9
= _____
Clicker Question:
Are these events independent?
A. Yes
B. No
Example
The probability that she will be enrolled in either French or
Spanish (or possibly both) is….
P(French) +P(Spanish) – P(both)
P(French or Spanish) = __________
0.6 + 0.5 – 0.2
= _______
0.9
= _____
Clicker Question:
Are these events mutually
exclusive?
A. Yes
B. No
Specific people vs. a random individual
As people, we constantly hear risks, statistics, and
probabilities
•
•
•
•
•
1 in 3.5 million planes crash
50% of marriages end in divorce
Acceptance rate at Penn State was 50.3% for
Fall 2014
Four-year graduation rate at Penn State is 66%
6-year graduation rate at Penn State 86%
Specific people vs. a random individual
As people, we constantly hear risks, statistics, and
probabilities
•
•
•
•
•
1 in 3.5 million planes crash
50% of marriages end in divorce
Acceptance rate at Penn State was 50.3% for
Fall 2014
Four-year graduation rate at Penn State is 66%
6-year graduation rate at Penn State 86%
Do these probabilities apply specifically to YOU?
Do these probabilities apply specifically to YOU?
• These statistics generally apply to the bigger
group – or to a person randomly selected from
that population.
• The proper language to communicate this is often
omitted.
• You should always understand that randomness
is part of the communication in these statistics,
even though it’s not explicitly mentioned.
Coincidences
Coincidence: a surprising concurrence of
events, perceived as meaningfully related, with
no apparent causal connection.
Examples
• Running into a friend in an unfamiliar city.
• Sharing a birthday with someone in your class.
• You meet someone who has a dog with your
name.
Coincidences
Coincidence: a surprising concurrence of
events, perceived as meaningfully related, with
no apparent causal connection.
Examples
• Running into a friend in an unfamiliar city.
• Sharing a birthday with someone in your
class.
• You meet someone who has a dog with
your name.
Many occurrences are
more common than we
suspect. Many are
unlikely for a specific
instance but do happen
when lots of instances
are possible.
What's the difference between these two statements?
"I'm confident that there is at least one set of matching
birthdays in this room”
2. "I'm confident that there is at least one person in this
room whose birthday matches my birthday"
1.
Which statement is more likely to be true?
How many possible pairs of people are eligible for
matching in each case? Assume 50 people are in
the room.
With 50 people in the room…
• There are 49 possible pairs with me.
• Pr (No match with my birthday) = (364/365)49 = 0.874
• There are 49+48+47+…+1 = 1225 total possible pairs.
• Pr (No match at all) = .030
and we can estimate by (364/365)1225 = .035
Shuffle two decks of cards.
• Stack the two decks side-by-side, face down next to each other.
• One by one, flip over one card from each deck.
• I bet I see at least one match. Do you want to bet against me?
Probability of no match:
• Probability of no match on 1st flip: 51/52
• Probability of no match on 2nd flip: 51/52
•…
• Probability of no match on 52nd flip: 51/52
These events are NOT independent; however, they are
APPROXIMATELY independent
2 on the 36th
because, say, whether
occurs
(57 −a match
46.4)
2.42
flip doesn't influence whether a match=
occurs
on
46.4
the 47th flip very strongly.
51 52
Thus, P(no match) ≈
= 0.364
52
Confusion of the inverse
Suppose that a particular disease affects 1% of those
who get tested for it. Also suppose that the test is 98%
accurate. What would you advise a patient who tests
positive if the test result were the only piece of
information?
True probability of disease: about 33%
Cancer testing: confusion of the inverse
Suppose we have a cancer test for a certain type of cancer.
Sensitivity of the test:
If you have cancer then the probability of a positive test
is .98. Pr(+ given you have C) = .98
Specificity of the test:
If you do not have cancer then the probability of a negative
test is .98. Pr(– given you do not have C) = .98
Base rate:
The percent of the population who has the cancer. This is
the probability that someone has C.
Suppose for our example it is 1%. Hence, Pr(C) = .01.
Percent table
+
Positive
–
Negative
Sensitivity
C
(Cancer)
.98
.02
.01
.02
.98
.99
Specificity
no C
(no
Cancer)
false positive
false negative
Suppose you go in for a test and it comes back positive.
What is the probability that you have cancer?
Base
Rate
Table of proportions (given):
+
–
Base rate
C
.98
.02
.01
no C
.02
.98
.99
Hypothetical table of counts:
Pr(C given a positive test result) = 98/296 = 33.1%
Probability review
Suppose we roll two fair dice, one red and one blue.
Let the event A be that we roll the same number on both dice.
Let the event B be that the sum of the two dice is even.
a. What is
P(A)?
b. What is
P(B)?
c. What is
P(A and B)?
a. What is
P(A or B)?
More probability review
Suppose that you flip a fair coin until the first occurrence of
tails.
What is the probability that the first “tails” occurs on the
third flip?
(A)
(B)
(C)
(D)
(E)
1/2
1/3
(1/2) × (1/2) × (1/2)
(1/2) × (1/4) × (1/8)
(1/2) + (1/4) + (1/8)
Gambler’s fallacy
• Long-term
probabilities should
apply in the short
term (false!)
• Random events
should be “selfcorrecting” (false!)
Example: a gambler who loses 48
times at a slot machine thinks that
they are about to win, since he
knows the slot machine pays big 1
in every 50 times in the long run.
Law of large numbers (this is true!)
If an event is repeated many times
independently with the same probability of
success each time, the long-run success
proportion will approach that probability.
• With independent events, knowing what has
happened tells you nothing about what will happen.
Misunderstanding this leads to the gambler’s fallacy,
also known as:
The “law of small numbers” (not a real thing),
which is that small samples will always be
representative of the population from which they
are drawn.
More on Gambler’s Fallacy
Suppose you flip four coins, keeping track of the results in
order.
• Which is more likely, HHHH or HTTH?
• Which is more likely, four total heads or
two total heads?
Note: These questions are not the same!
One of these questions is often mistakenly
answered due to belief in the "Law of small
numbers" (also known as the Gambler's Fallacy).
More probability review
Suppose that you roll a fair 6-sided die until the first
occurrence of a 4.
What is the probability that the first 4 occurs on the third
roll?
(A)
(B)
(C)
(D)
(E)
1/6
5/6
(5/6) × (5/6) × (1/6)
(1/6) × (1/6) × (1/6)
(1/6) + (1/6) + (1/6)
If you understand today’s lecture…
• 7.83, 7.84, 7.85, 7.88, 7.89, 7.91, 7.92
Recall Objectives:
• Identify complementary events and handle probability calculations
• Identify mutually exclusive events and handle probability calculations
• Identify independent events and handle probability calculations
• Understand that conditioning on a particular individual can change risks
• Develop a sense for why seeming coincidences occur frequently
• Identify, and resist the temptation to fall for, the “gambler’s fallacy”
• Understand a common situation where confusion of the inverse occurs