Day 02, MAT115-006, Fall 2010

Last Time

Next Time

Roll Call
Announcements:
- Please hand in your class agreements
- Also hand in your first assignment.
Now, back to the fun and games!
- We've been talking about probability: just what do we mean by "The probability of an event is ...."?
  - If I flip a fair coin, what's the probability that I'll get heads?
  - Does a probability of 1/2 mean that if I get heads this time, I have to get tails the next time?
  - If I roll a die, what's the probability that I'll get a 5?
  - ${Probability(DesiredOutcomes)=\frac{DesiredOutcomes}{TotalOutcomesPossible}$
- Let's investigate the "Let's Make a Deal!" game mathematically:
  1. A reminder of the game:
    There are three doors, behind which are two donkeys and a car.
    You choose one door: Monty Hall then shows you a door behind which appears a donkey, then offers you the chance to either "stick" with the door you chose at first, or "switch" for the third door.
    Do you stick or switch? How can we decide?
  2. It turns out that there is an "optimal strategy". Your mission: determine what strategy is best. In this class, after a short discussion with a neighbor or two, we reached the following tentative conclusion:
    
    Stick Switch Doesn't Matter
    
    8 21 5
    
    Votes for Best Strategy, upon careful reflection...?
  3. Maybe we can simulate the game.... Let's break into groups to play:
    - Three cards represent the three doors; choose one card to be the "car", and the other two to be the donkeys.
    - Try 10 (or more) plays using the sticking strategy.
    - Tally up the successes of each strategy (realizing that if sticking wins, switching would have lost, and vice versa).
    - Is one strategy better? If so, by how much? "What are the odds?"
    - Then we'll combine the class's data, to "see what the data says".
    - Now: Can you justify/explain your answer?
  4. To give you more confidence in the answer (and perhaps to learn more!), we may consider extending the game: what if we have ten doors, and, after you pick your door, Monty shows you eight doors with donkeys behind them. Would you switch for the remaining door? How much better is one strategy over the other?
  5. Our goal in this class is "more than just mathematics" -- there are "Lessons for life" that we should focus on. For this problem, I'd like to mention the following:
    1. Just do it.
    2. Make mistakes and fail, but never give up.
    3. Explore the consequences of new ideas.
    4. Look for patterns and similarities.
- Now let's consider the birthday problem (Idea #33 in the text (p. 132) -- you were to have read this section for today). This is also a probability problem.
  - First off, what do you think of the style of the writing?
  - Did you notice the timeline?
  - I have to mention a "weird coincidence": a Bob Dylan concert is mentioned in this section, and just yesterday I heard on NPR a news story about a Bob Dylan concert in San Francisco. It was interesting because ticket sales ($60 each) started at 5:30; the show was at 8:00; and you had to go straight in to the show upon buying your tickets -- so there would be no scalping (and no internet buy-outs, etc.).
    What's the probability that, just as we discuss this issue, we'd see such a coincidence?
  - Now, back to birthdays.... How does our author approach the solution to the question "what's the likelihood that two randomly selected people have the same birthday?"
    1. What about leap years?
    2. Cases: two people
      - Probability that they have the same birthday
      - 1 - (Probability that they have different birthdays)
      Notice how our author deftly replaces one mathematical problem with another which is easier to handle. This is a mathematician's job!
    3. Now, what happens as we add people?
      - Probability that any pair has the same birthday
      - 1 - (Probability that they all have different birthdays)
    4. Birthday probability calculator:
- Having answered "the birthday problem", what does our author consider next? "The birthday problem has been generalized in many ways...." How?
- "Lessons for life" relevant to this exercise:
  1. Just do it.
  2. Make mistakes and fail, but never give up.
  3. Break a difficult problem into easier ones.
  4. Don't be a turkey -- be a dog! (this one may require a little explanation....)
  5. Explore the consequences of new ideas.
  6. Look for patterns and similarities.
  7. Understand simple things deeply.
  8. Examine issues from several points of view.
Now, if we have any time left, let's consider our homework problems. Which of those would be interesting to consider together?

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