1. Statistics 200 Objectives:
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.

2. 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)

3. 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 Pr(+ given you have C) = .98
Specificity of the test:
If you do not have cancer then the probability of a negative
test is 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.

4. Table of proportions (given):+ –
Base rate C
.98
.02
.01
no C
.99
Hypothetical table of counts: + – C 98 2
100
no C
198
9702
9,900
296
9704
10,000
Pr(C given a positive test result) = 98/296 = 33.1%

5. Gambler’s fallacy
Long-term probabilities should apply in the short term (false!)
Random events should be “self-correcting” (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.

6. 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.

7. 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).

8. 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?
1/6
5/6
(5/6) × (5/6) × (1/6)
(1/6) × (1/6) × (1/6)
(1/6) + (1/6) + (1/6)

9. Still more probability review
Consider the diagram.
Which number should go in the “??” Box?
A. 1.0
B. 0.2
C. 0.1
D. 0.9 B A ?? .4 .3 .1

10. 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.

11. Let X = number of heads X = 3 X = 0 X = 1 X = 2 X = 2 X = 2 X = 1
Capital letters like X or Y denote random variables
Let X = number of heads
X = 3
X = 0
X = 1
X = 2
X = 2
X = 2
X = 1
X = 1
Later, we’ll assign a probability to each possible value of X.

12. Examples: Discrete Random Variables
Possible values of X
Flip 3 coins
X= # heads
0, 1, 2, 3
2, 3, …, 11, 12
Roll 2 dice
X= sum of dots
Survive Monday
X= # cups of coffee
0, 1, 2, …, 10, …

13. 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.

14. 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?

15. Continuous Random Variable
Assumes a range of values covering an interval. _____________.
May be limited by instrument’s accuracy / decimal points, but still continuous.
Find probabilities using a probability density function, which is a curve.
Calculate probabilities by finding the area under the curve.
is this area
• We can’t find probabilities for exact outcomes.
• For example: P(X = 2) = 0.
• Instead we can find probabilities for a range of values.

16. Coin example: build pdf for X = # heads from 3 coin tossesk
P(X=k)
__ / 8 1 2 3
Total 1

17. Coin example: build pdf for X = # heads from 3 coin tossesk
P(X=k)
__ / 8 1 2 3
Total 1 3

18. Coin example: build pdf for X = # heads from 3 coin tossesk
P(X=k)
__ / 8 1 2 3
Total 1 3 3

19. Coin example: build pdf for X = # heads from 3 coin tossesk
P(X=k)
__ / 8 1 2 3
Total 1 3 3 1

20. Example 2: An example of a probability distribution function (pdf)
Let X = number of courses per semester taught by PSU faculty k
P(X = k)
0.1 1
0.3 2
0.4 3
0.2
Total
1.0
Why is this table a pdf?

21. X = number of courses/semester taught by PSU faculty
What is the probability that X is:
exactly 2 classes?
at most 2 classes?
at least 1 class?
more than 1 class?
Note: need to be careful at the borders k
P(X = k)
0.1 1
0.3 2
0.4 3
0.2
Total
1.0

22. X = number of courses/semester taught by PSU faculty
What is the probability that X is:
exactly 2 classes?
at most 2 classes?
at least 1 class?
more than 1 class?
Note: need to be careful at the borders k
P(X = k)
0.1 1
0.3 2
0.4 3
0.2
Total
1.0

23. X = number of courses/semester taught by PSU faculty
What is the probability that X is:
exactly 2 classes?
at most 2 classes?
at least 1 class?
more than 1 class?
Note: need to be careful at the borders k
P(X = k)
0.1 1
0.3 2
0.4 3
0.2
Total
1.0

24. X = number of courses/semester taught by PSU faculty
What is the probability that X is:
exactly 2 classes?
at most 2 classes?
at least 1 class?
more than 1 class?
Note: need to be careful at the borders k
P(X = k)
0.1 1
0.3 2
0.4 3
0.2
Total
1.0

25. Section 8.3: Expected Value for Any Discrete Random Variable
Expected Value of X
Average value in the long run.
notation
µ:`mu’
In other words, mean from an imagined infinite number of observations.
When calculated using a pdf:
a weighted average

26. Faculty Example: Calculate the Expected Value E(X)
Expected Value = Sum of “value × probability” over all possible values
Faculty Example: Calculate the Expected Value E(X)
X = # courses/semester taught by PSU faculty k
P(X = k)
0.1 1
0.3 2
0.4 3
0.2
Total
1.0
E(X) = µ =
Interpretation:
The average is 1.7 classes for this population
0×(.1) + 1×(.3) + 2×(.4) + 3×(.2)
= 1.7

27. Expected Value Application
Probability that Mary wins a game is 0.01.
If she wins: gets $100
If she loses: pays $5.
X = the amount of money Mary will gain
Outcome
X Value
Probability
Total NA
0.01
Win
100
Lose -5
0.99
1.00
E(X) =
(100)×(.01) + (–5)×(.99)
= 1 – 4.95
Questions:
• Variable X is: discrete continuous
• Set up pdf
• What is the expected gain for Mary?
= –3.95

28. 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.