- You have an assignment due Friday -- see the assignment page, and make sure to keep up!
As you do this homework, remember our "lessons for life":
- Just do it. Jump in. Make mistakes and fail, but never give up.
- Keep an open mind.
- Understand simple things deeply.
- Break a difficult problem into easier ones.
- Look for patterns and similarities.
- Explore the consequences of new ideas. Generalize.
- Examine issues from several points of view.
Don't be a turkey -- be a dog! (this one may require a little explanation....)
Please don't just go out on-line and download someone else's solutions!
- You also have a new reading assignment: read the birthday problem
(Idea #33 in the text -- p. 132) for next time.
- First off, what do you think of the style of the writing?
- Did you notice the timeline?
- Remember our "Question of the day" from last time:
What's the probability that two people in this room have the same birthday? So I've got $10 that says that there are two people in here who have the same birthday. How much will you bet against me?
"Odds" is an equivalent way of thinking about this probability: how much would you bet against my bet of $10 that there will be two people in this room with the same birthday?
Even money (odds)? If so, then you would be saying that the probability of there being two people with the same birthday is 1/2.
- Let's think about some simpler problems than the birthday
problem. This problem (and the other problems we'll
consider today) is all about probability: just what do
we mean by "The probability of an event is ...."?
- What's the probability that a die comes up 5? 6? 5 or 6?
(This is the focus of your first reading.)
- Is there such a thing as "being due"? If my probability of making
a freethrow in basketball is 1/2, and I miss this shot, will I (of
necessity) make the next? If I miss three in a row, am I due?
- Generally, if the situation is "fair" (meaning that all
events have the same chance of happening), then
- What's the probability that a die comes up 5? 6? 5 or 6?
(This is the focus of your first reading.)
- So let's do some calculations:
- Assuming a fair six-sided die, let's calculate the probability that a die comes up 5, 6, and (5 or 6).
- Let's look at another problem: what's the probability of tossing a coin three times and getting exactly two heads?
- Now, it's not always easy to figure out the theoretical probabilities (as we did in the last two examples).
What's the probability that a 78-year-old white female in Wood County, Ohio will survive another year?
That's the kind of question an insurance adjuster is concerned about, as am I: that's my mom!
An approximation can be obtained by looking at records for Wood County for the last year:
So let's do an experiment to see how close we get to our calculated value...
- Now your reading told you about the game of craps. We were
able to use a table to 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 for a roll of two dice. Let's do a few more calculations:
-
- Let's consider the birthday problem (Idea #33 in the text (p. 132)
-- read this section for next time).
- What about leap years? Forget 'em!
- Let's suppose there are 37 people in the room. How many total ways could we 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.
- Birthday probability calculator: