Day 33, MAT115

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Announcements:
- Your (rather sad) quizzes are returned. I'd suggest that some of you need to get up to speed by actually doing the reading.... It was a total of 14 pages.
  A key
- I've put another probability reading on the assignment page: this chapter about the birthday problem from Tony Crilly's "50 Mathematical Ideas You Really Need to Know".)
- Your quiz this Friday will be on probability.
Today's Question of the Day (recycled!):

What's the probability that two people in this room have the same birthday?
What are legitimate values for probabilities? Numbers between 0 and 1.
Let's take a moment to compare chance, probability, and odds. All are used in "games of chance".
- If the chance of rain today is 50%, then on 1 out of 2 days like today you'd expect rain.
- Equivalently the probability of rain is 1/2, or .5
- Another way of saying it is that the odds are 50/50 (or 1 to 1) that it will rain today.
Let's adjust the example a little.
- If the chance of rain today is 20%, then on 1 out of 5 days like today you'd expect rain.
- Equivalently the probability of rain is 1/5, or .20
- Another way of saying it is that the odds are 20/80 (or 1 to 4) that it will rain today.
We could work within any one of these systems -- chance, probability, or odds, but we'll work with probability.

Now your reading in Crilly's book talks about the game of craps. We are able to use a table to represent the universe of possible outcomes, from which we can compute the probability of a 7 or 11:

	1	2	3	4	5	6
1	(1, 1)	(1, 2)	(1, 3)	(1, 4)	(1, 5)	(1, 6)
2	(2, 1)	(2, 2)	(2, 3)	(2, 4)	(2, 5)	(2, 6)
3	(3, 1)	(3, 2)	(3, 3)	(3, 4)	(3, 5)	(3, 6)
4	(4, 1)	(4, 2)	(4, 3)	(4, 4)	(4, 5)	(4, 6)
5	(5, 1)	(5, 2)	(5, 3)	(5, 4)	(5, 5)	(5, 6)
6	(6, 1)	(6, 2)	(6, 3)	(6, 4)	(6, 5)	(6, 6)

The sample space (or "universe") for a roll of two dice.

Let's do a few more calculations:

${P(doubles)}$
${P(sum\le{7})}$

Let's reconsider the birthday problem:

What about leap years? (Forget 'em! Mathematicians like to simplify things down to make them easier to do, and then ask how badly does that mess up our problem....)
Suppose there are 30 people in a room. How many total ways could they have birthdays? Since each person makes their "choice" independently, we multiply.
An event is either a success (two people share a birthday) or a failure (no common birthdays). It's actually easier to compute the number of failures. We use the idea of complementary events:
If two events A and B are complementary, then the universe divides into two parts: events of type A, or events of type B.
We sometimes denote B as A' -- meaning A-not, or "not A". Every event is either A, or not A (B).
That is, it's a binary world!
Birthday probability calculator:

Now let's take a look at our reading from The Joy of X (chapter 23 -- Chances Are)

Well, right away our author does something bone-headed -- he says that "...the probability that your friend will forget to water it is 30 percent." That's the chance that your friend will forget to water it, right? The probability is .30.
Let's start by considering the universe. We'll represent it in a slightly different way, however, using a table. The first problem to consider is the problem of the plants, living or dead, and the watering (done or not). Because there are these two ways of looking at the situation of our plants and their waterings, our universe will be divided into two separate binary worlds....

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