Day 04, MAT115

Last Time

Next Time

Announcements:
- Your first assignment is due.
- I'll collect your votes for the top three topics from our textbook. If you wish to vote, I'll hand you a card. Otherwise, you may abstain.
- Once again, you have a reading assignment for next time.
Question of the day: What else can we learn from Let's Make a Deal?
The Textbook
You were to read your first section ("My Tub Runneth Over") from our textbook. I wanted you to read that essay to get a feeling for Strogatz's writing. I think that it's a good introduction to "the word problem" (aka "story problem") as well.

What did you think? Is it at all entertaining?
What did you think of the solutions to the
- tub problem?
- fence problem?
In what sense are we inclined to be turkeys for each of those? How did we become dogs?
There were a couple of great points in this essay, which may apply to you:
- "Three men can paint three fences in three hours."
  When I heard the fence problem question: "How long would it take one man to paint one fence?" I thought of another Monty: Monty Python and the Holy Grail (can you guess which part?)
  If you asked "which man?" in the fence problem, then you're inclined to make problems realistic, which makes them more difficult to solve.
  We often solve a simpler problem because we can.
- Strogatz said something else interesting: word problems "...give us practice in being mindful."
Let's get mindfully back to the business of our "Fun and Games" (probability):
Last time we learned that, in the Monty Hall problem, switching beats sticking, by 2 to 1 (that's an odds). In terms of probability,

${P(winning\ by\ switching)={\frac{2}{3}}}$

${P(winning\ by\ sticking)={\frac{1}{3}}}$
Or is there still anyone out there who isn't sure? Because I want everyone convinced by the end of the day....
Sticking and switching are complementary strategies. One strategy will win, and one strategy will lose. Hence,

P(win by sticking)+P(win by switching)=1
In terms of fair bets, if I were to bet $2 on switching, you'd bet $1 on sticking, and that would be fair.
The bets are in the same ratio as the probabilities:

${\frac{2}{1}={\frac{P(winning\ by\ switching)} {P(winning\ by\ sticking)}}}$
This ratio is also known as the odds of my winning by switching.
So one thing that we've learned is that there are different ways of talking about the "likelihood" of winning:
- likelihood
- chance (percentage)
- odds (ratio)
- probability (number between 0 and 1)
- First of all, we should understand simple things deeply. Let's change the game. The first time we change it, we'll make a modest change. But we seek to "explore the consequences of new ideas. Generalize."
  One way to make the strategy of switching more evident is to increase the number of doors. What if we play the game with
  - four doors? (Monty opens two)
  - ten doors? (Monty opens eight)
  - ten thousand doors? (a lottery)
  By changing the problem, and considering even more complex problems, we learn more about the simple problem.
- Now: let's really change the game: let's look at two variations of the Monty Hall problem (there are a million), and decide what we should do.
  1. So what if we have five doors, hiding two cars and three donkeys. Suppose you can pick two doors, and receive a car if either door you pick hides a car.
    Monty Hall looks at the other three doors, and shows one that has a donkey behind it. Then he offers you the two other doors for your two. Do you switch?
    It's one thing to make a choice -- it's another to make an informed choice. Discuss the situation with your neighbor.
    What is the probability that you win a car by sticking, versus the probability that you win a car by switching?
    Sometimes probability comes down to counting. But your strategy will have to change....
    - What's the universe? Can you use a tree to construct it?
    - Can you simply the universe?
    - Which outcomes result in a win for sticking?
    - Which outcomes result in a win for switching?
    - What are the probabilities?
    - Which strategy is better?
    - What is dramatically different about these probabilities?
  2. Now suppose that you receive a car only if you find BOTH cars behind your two doors.
    How does the problem change? (There's something dramatically new here!)
    - What's the universe? (Did it change?)
    - Which outcomes result in a win for sticking?
    - Which outcomes result in a win for switching?
    - What are the probabilities?
    - What is dramatically different about these probabilities?

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