Download PowerPoint

STATISTICS 200
Lecture #13
Tuesday, October 4, 2016
Textbook: Sections 7.3, 7.4, 8.1, 8.2, 8.3
Objectives:
• Identify, and resist the temptation to fall for, the “gambler’s
fallacy”
• Define “random variable” and identify the difference between
discrete and continuous.
• Use and recognize probability distribution functions (for
discrete) or density functions (for continuous)
• Calculate and interpret the expected value of a discrete
random variable.
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 given this information : about 33%
Confusion between
P(disease present given positive test)
and
P(positive test given disease present)
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.
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%
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
he is 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)
Still more probability review
Consider the
diagram.
B
A
.3
.1
??
.4
Which number should
go in the “??” Box?
A. 1.0
B. 0.2
C. 0.1
D. 0.9
What is a random variable?
A random variable is a rule for
assigning a numerical value to
each outcome of a probability
experiment.
Example: The number
of heads that appear if I
toss a nickel, a dime,
and a quarter.
Capital letters like
X or Y denote
random variables
Let X = number of
heads
X=1
X=2
X=3
X=0
X=2
X=2
X=1
X=1
Later, we’ll assign a probability to each possible value of X.
Examples: Discrete Random Variables
Possible values of X
Flip 3 coins
X= # heads
0, 1, 2, 3
Roll 2 dice
X= sum of dots
2, 3, …, 11, 12
Survive Monday
X= # cups of coffee
0, 1, 2, …, 10, …
Discrete Random Variables
Discrete list of distinct values.
Typically used when a variable is integer-valued
without too many possible choices. choices
pdf (Probability distribution
function)
For example: P(X = 2) = 0.278
If X is the sum of two fair dice when rolled.
Examples: Continuous Random Variables
Possible values of X
Catch the R bus
X= time waiting
Between 0 and 20, ideally
Weigh male firefighter
X= his weight
Between 150 and 250
Run obstacle course
X= time to finish
0 to infinity?
Continuous Random Variable
Assumes a range of values covering an interval.
_____________.
May be limited by instrument’s accuracy / decimal points,
but still continuous.
is this
area
Find probabilities using a
probability density function,
which is a curve.
Calculate probabilities by
finding the area under the
curve.
• We can’t find probabilities for exact outcomes.
• For example: P(X = 2) = 0.
• Instead we can find probabilities for a range of
values.
Coin example: build pdf for X = # heads from 3 coin tosses
k
P(X=k)
0
1 /8
__
1
__ / 8
2
__ / 8
3
__ / 8
Total
1
Coin example: build pdf for X = # heads from 3 coin tosses
k
P(X=k)
0
1 /8
__
1
3 /8
__
2
__ / 8
3
__ / 8
Total
1
Coin example: build pdf for X = # heads from 3 coin tosses
k
P(X=k)
0
1 /8
__
1
3 /8
__
2
3 /8
__
3
__ / 8
Total
1
Coin example: build pdf for X = # heads from 3 coin tosses
k
P(X=k)
0
1 /8
__
1
3 /8
__
2
3 /8
__
3
1 /8
__
Total
1
Example 2: An example of a probability
distribution function (pdf)
 Let X = number of
courses per semester
taught by PSU faculty
k
0
1
2
3
Total
P(X = k)
0.1
0.3
0.4
0.2
1.0
Why is this table a pdf?
X = number of courses/semester taught by PSU faculty
What is the probability that X is:
• exactly 2 classes?
k
P(X = k)
0
0.1
1
0.3
2
0.4
3
0.2
Total
1.0
• at most 2 classes?
• at least 1 class?
• more than 1 class?
Note: need to be careful at the borders
X = number of courses/semester taught by PSU faculty
What is the probability that X is:
• exactly 2 classes?
k
P(X = k)
0
0.1
1
0.3
2
0.4
3
0.2
Total
1.0
• at most 2 classes?
• at least 1 class?
• more than 1 class?
Note: need to be careful at the borders
X = number of courses/semester taught by PSU faculty
What is the probability that X is:
• exactly 2 classes?
k
P(X = k)
0
0.1
1
0.3
2
0.4
3
0.2
Total
1.0
• at most 2 classes?
• at least 1 class?
• more than 1 class?
Note: need to be careful at the borders
X = number of courses/semester taught by PSU faculty
What is the probability that X is:
• exactly 2 classes?
k
P(X = k)
0
0.1
1
0.3
2
0.4
3
0.2
Total
1.0
• at most 2 classes?
• at least 1 class?
• more than 1 class?
Note: need to be careful at the borders
25
Section 8.3: Expected Value for Any
Discrete Random Variable
Expected Value of X
notation
µ:`mu’
Average value in
the long run.
In other words, mean
from an imagined infinite
number of observations.
When calculated using a pdf:
a weighted average
26
Expected Value = Sum of “value × probability”
over all possible values
Faculty Example: Calculate the Expected
Value E(X)
k
P(X = k)
0
0.1
1
0.3
2
0.4
3
0.2
Total
1.0
X = # courses/semester taught by
PSU faculty
 E(X) = µ =
0×(.1) + 1×(.3) + 2×(.4) +
3×(.2)
= 1.7
Interpretation:
• The average is 1.7 classes for this
population
27
Expected Value Application
• Probability that Mary
Outcome
Win
wins a game is 0.01.
• If she wins:
Lose
gets $100
Total
• If she loses:
pays $5.
• X = the amount of
E(X) =
money Mary will gain
Questions:
X Value
100
-5
NA
Probability
0.01
0.99
1.00
(100)×(.01) + (–5)×(.99
= 1 – 4.95
= –3.95
continuous
• Variable X is: discrete
• Set up pdf
• What is the expected gain for Mary?
If you understand today’s lecture…
8.1, 8.7, 8.9, 8.15ab, 8.17, 8.21, 8.25, 8.29,
8.33, 8.37
Objectives:
• Identify, and resist the temptation to fall for, the “gambler’s
fallacy”
• Define “random variable” and identify the difference between
discrete and continuous.
• Use and recognize probability distribution functions (for
discrete) or density functions (for continuous)
• Calculate and interpret the expected value of a discrete
random variable.