- Reminder: you have an assignment due Friday.
- New assignment (by Friday): check out my on-line description of The Birthday Problem (and use it to figure out if it's going to rain by Saturday). Just read it over, and come with any questions.
- Any last comments on homework from Monday?
- Remember: the best solution is the one that assumes the least.
- Some assumptions are necessary, e.g.
- that the monsters can communicate with us;
- that they know the pathway to heaven and hell;
- that they're not tricky!
- Worst "answer": assume that you have a magic genie with you, who knows all the answers to all the tricky logic problems. You're done.
- Okay answer: assume that you have monsters who
know the behaviors of other monsters (e.g. they know
which ones are liars and which are truth-tellers): then
If I were to ask THE OTHER MONSTER to point to the path leading to heaven,
which way would THE OTHER MONSTER point? - Best: assumes nothing about monsters, other than
that they live up to their billing as either ultimate
liars or truth-tellers:
If I were to ask YOU to point to the path leading to heaven,
which way would YOU point?
- Some assumptions are necessary, e.g.
- Remember: the best solution is the one that assumes the least.
- Here's our "Question of the day" (one more probability problem):
In the game of "Let's make a deal!" is it better to stick or switch? - Let's consider that famous mathematical problem called "Let's make a deal!"
- The rules:
- There are three doors, behind which are two donkeys and a car.
- You choose one: 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?
- It turns out that there is an "optimal strategy". Your mission: determine what strategy is best.
- We should be able to do this theoretically:
- What's the sample space?
- What's your probability of success given that you stick?
- What's your probability of success given that you switch?
- Why don't we simulate the game.... Let's break into
groups to play, and see if our theoretical answer is correct:
- 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?
- Is it possible that by restricting the game, or by extending the
game, we might learn more?
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?
- The rules:
- Let's begin looking at the history and make-up of numbers.
- Of course we'll start with the simplest numbers, which are the natural numbers:
- The natural numbers are the counting numbers: 1, 2, 3, ....
- All natural numbers are interesting, and we can prove it!
- Is 1 interesting?
- Is 2 interesting?
- Suppose there were a first uninteresting natural number....
- The number 1729 is especially interesting: the story of
Ramanujan and Hardy
This story illustrates that we can break down numbers in interesting ways.
- The natural numbers are the counting numbers: 1, 2, 3, ....
- Now, zero didn't make it right into the mix: it just confused the
heck out of everyone. Next came
- "rational numbers" -- ratios of natural numbers
- "irrational numbers" -- e.g. the side length of a square with sides of length 1
- negative numbers
- But then finally zero had its day -- what's its history?
- Our author says that "Mathematics could not function without zero." (p. 7) How do we know that he's wrong?
- What civilizations were the first to include zero as a number?
- Prime Numbers: breaking numbers down by multiplication
- What is a prime number?
- A natural number that can be divided evenly (that is, without remainder) by only two distinct natural numbers: 1 and itself.
- NOTE: when mathematicians say "distinct" natural numbers, they mean that they need to be different; and so the number 1 is not prime.
- We can find the primes using the Sieve of Eratosthenes (here's an animation)
- Prime Factorization Theorem:
- Every natural number greater than 1 is either prime, or it
can be expressed as a product of prime numbers (in one
and only one way -- order of the product aside).
- Examples:
- 42=2*21=2*3*7
- 8=2*2*2 (prime factors can repeat)
- 1729 = 7*247 = 7*13*19
- If you can't factor by any prime less than or equal to the number's square root, then the number is prime.
- 127: its square root is approximately 11.27. We don't need to go past 11 to find its factors.
- The general procedure (and one that we'll see
repeated time and again in mathematics):
do it once, then do it again, and again, and again.... until done (or tired). I'll call this out sometime as "do it again...."
- Upshot: We can "decompose" (or factor) natural numbers using prime numbers and products.
- Every natural number greater than 1 is either prime, or it
can be expressed as a product of prime numbers (in one
and only one way -- order of the product aside).
- What is a prime number?
- Of course we'll start with the simplest numbers, which are the natural numbers:
