1. [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?

  2. [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

  3. [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?

  4. [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

  5. [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)

  6. [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

  7. [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

  8. [The Secretary Problem]
    Pick the largest number
    The Secretary Problem (Wikipedia)

  9. [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?

  10. 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. 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?

  11. 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