- [Birthday Problem]
In a group of 5 people, how likely is it that 2 of them have the
"same" birthday?
What about a group of 25 (or 50) people?
- [Bus problem] Two different bus lines run on a road. Each comes
every 10 minutes, but their schedules are not coordinated. Both of
them stop at your destination. You use this every day. On the
average how long will you need to wait for a bus?
Solution
- [Shadyrest Hospital]
Shadyrest Hospital draws its patients from a rural area that has
twelve thousand elderly residents. The probability that any one of the
twelve thousand will have a heart attack on a given day and will need
to be connected to a special cardiac monitoring machine has been
estimated to be one in eight thousand.
Currently the hospital has three such machines. What is the
probability that the equipment will be inadequate to meet tomorrow's
emergencies?
- [Which Door?]
Suppose you're on a game show, and you're given the choice of three
doors: Behind one door is a car; behind the others, goats. You pick a
door, say No. 1, and the host, who knows what's behind the other
doors, opens another door, say No. 3, which has a goat. He then says
to you, 'Do you want to pick door No. 2?' Is it to your advantage to
take the switch?
Discussion
- [Cancer Test]
Suppose that you undergo a medical test for a relatively rare
cancer. Your doctor tells you that, according to surveys by medical
statisticians, the cancer has an incidence of 1% among the general
population. Thus, before you take the test, and in the absence of any
other evidence, your best estimate of your likelihood of having the
cancer is 1 in 100, i.e. a probability of 0.01. Then you take the
test. Extensive trials have shown that the reliability of the test is
79%. More precisely, although the test does not fail to detect the
cancer when it is present, it gives a positive result in 21% of the
cases where no cancer is present -- what is known as a "false
positive."
When you are tested, the test produces a positive
diagnosis.
The question is: Given the result of the test, what is the
probability that you have the cancer?
See:
Devlin: Bayes,
Bayes Formulas,
A Tree Diagram.
Also:
Grinstead and Snell, Chapter 4 (pdf)
- [Weighing the evidence. (Amos Tversky and Daniel Kahneman)]
A certain town has two taxi companies, Blue Cabs and Black Cabs. Blue
Cabs has 15 taxis, Black Cabs has 85.
Late one night, there is a hit-and-run accident involving a
taxi. All of the town's 100 taxis 0were on the streets at the time of
the accident. A witness sees the accident and claims that a blue taxi
was involved.
At the request of the police, the witness undergoes a vision test
under conditions similar to the those on the night in
question. Presented repeatedly with a blue taxi and a black taxi, in
random order, he shows he can successfully identify the color of the
taxi 4 times out of 5. (The remaining 1/5 of the time, he
misidentifies a blue taxi as black or a black taxi as blue.)
If you were investigating the case, which company would you think is
most likely to have been involved in the accident?
Tversky-Kahneman (Devlin)
See also the superb book, Kahneman, D. "Thinking, Fast and Slow,"
Farrar, Straus, Giroux; New York (2011) -- and --
Vanity Fair: Kahneman Article,
Kahneman Quiz
- [Markov Chains]
Introduction,
Grinstead and Snell, Chapter 11 (pdf) (Caution: Their
transition matrices are the transpose of ours.)
Brin-Page: How Google Works, 1998,
"The Page Rank Citation Ranking: Bringing Order to the Web",
Page, L., Brin, S., Motwani, R., & Winograd, T. (1999),
Beyan & Liese: "The $25,000,000,000 Eigenvector"
Indexing an article (inverse document frequency): Karen Spark Jones
Sports Ranking
Kendal-Wei Ranking
Ivy League Basketball 2002
College Football NYT 2001
College Football NYT 2012
Ranking Sports Teams Using the Perron-Frobenius Theorem
College Football Rankings by Computers, 2002
- [The Secretary Problem]
Pick the largest number
The Secretary Problem (Wikipedia)
- [Making a triangle]
You have 3 sticks of length x, y, and z, respectively with x+y+z=1.
What is the probability you can assemble them into a triangle?
- Understand the meaning of "average" of a data sample. In what
sense is the usual "mean" a good measurement of the "average"?
Simpson's Paradox: "In a certain hospital, there are two
surgeons. Surgeon A operates on 100 patients, and 95 survive. Surgeon
B operates on 80 patients and 72 survive. We are considering having
surgery performed in this hospital and living through the operation is
something that is important. We want to choose the better of the two
surgeons.
We look at the data and use it to calculate what percentage of surgeon
A's patients survived their operations and compare it to the survival
rate of the patients of surgeon B.
- Surgeon A: 95 patients out of 100 survived, so 95/100 = 95% of
them survived.
- Surgeon B: 72 patients out of 80 survived, so 72/80 = 90% of
them survived.
From this analysis, which surgeon should we choose to treat us? It
would seem that surgeon A is the safer bet. But is this really true?
See
Simpson's Paradox
Sex Bias in
Graduate Admissions?
- Independent Random Variables
Prob(A ∩ B) = Prob(A)Prob(B).
Examples: 2 dice (4,4), cards (pair of Kings)
California vs Collins, 1964 Los Angeles robbery