Probability Practice Test and Key
November 30, 2007
I order to give you a little more practice before our test on MONDAY (12/3). Here are a practice these and key. Email me if you have any questions or issues.
Practice Test Practice Test Key
Mr Page
Disjoint and Independence Examples
November 30, 2007
I was trolling on the AP Stats List serve and I came upon some good descriptive examples that may help you better understand Disjoint and Independence. These are from a Statistician David Boch:
Disjoint events cannot both occur at the same time. If one
happens, the other does not – guaranteed.
Independent events must be able to occur at the same time. If one
happens, it has no influence on the other whatsoever. The occurrence of one
provides no information about the other.
I draw a card from a deck. You have to bet on whether or not
it’s red. You have a 50-50 chance of being right. Now suppose that before you bet, I peek at
the card and tell you it’s a spade. Does that help you place your bet? Of
course! Note that the events “red” and “spade” are disjoint
- there are no red spades. Because of
that, knowing the card is a spade is a dead giveaway; it tells you for sure
that the card can’t be red. The probability the card is red has changed from
1/2 to 0. These two events are NOT independent.
OK, let’s play again. This time I peek at the card and tell
you it’s an ace. Does that help you place your bet? Before you knew this, the
probability the card is red was 26/52 = 1/2. Knowing it’s an ace, the
probability it’s red is 2/4 = 1/2. No help whatsoever – the probability has not
changed. These two events ARE independent (and not disjoint). P(red | ace) =
P(red) — that’s the very definition of independence:
the occurrence of “ace” has no effect on the probability of
“red”.
Initially students sometimes can’t tell the difference
between independent and disjoint, but that’s not the only potential problem
here. Once it becomes clear that disjoint events cannot be independent
(disjoint -> not independent), sometimes they think that the inverse is true
(not disjoint -> independent). Nope. Events that aren’t disjoint may or may
not be independent. Most of the students have taken biology, and that
background provides a nice example: colorblindness occurs more often in men
than in women. A woman can be colorblind, so the events are not disjoint. But
the probability is much lower, so sex and colorblindness are not independent
either
Independence
should be thought of in terms of conditional probability: A and B are
independent if P(B | A) = P(B). This
clearly says the probability of B remains the same even if we know that A has
occurred, this is the essence of independence.
Probability Worksheet Keys
November 30, 2007
Here are the keys for the 2 Stat worksheets
Probability Worksheet (1st One)
Probability II Worksheet (2nd One)
We will review the second one tomorrow (Friday 11/29). Make sure you understand these problems.
Mr. P
Transformation Quiz Homework
November 25, 2007
For those of you who did not do well on the transformation quiz, please print out the following worksheet and complete. If you did OK on the quiz, you may do them as extra credit. Make sure you understand how to do logs!!! If you can not solve log or transformation problems, see me at lunch or after school on Monday or Tuesday (11/26-27).
Thanksgiving Assignment
November 23, 2007
Happy Thanksgiving! I hope all of you
are enjoying the Holiday and time off. So far, the transformation
“quests” are very good, and I am pleased with most of your efforts and
understanding!! As a special Mr. Page bonus, I only want you to read
the second half of your “probability pamphlets” and 6.3 Probability
Rules. See you all on Monday. I will have your grades posted later in
the weekend!
Mr. Page
Transformations
November 7, 2007
Transformations can be ugly, scary and monstrous at
times. Don’t let them get you down. Make sure you know how to
manipulate logs (or lns). This is important and makes the math less
daunting.
The process of transformation is really fitting a curve to curved data. However, since the data is curved, we would have no way to determine if our “curve” is a good one. With linear data we can tell if our regression line is a good model for the data by the residual plot. A good uniform scater of points would represent a good linear model. Therefore, the goal of transformations is to
transform nonlinear
data into a linear data. If you check the residuals of the transformed data and the data is uniform, then your transformed data is a good linear model, and your untransformed regression line (curve) is a good model of the original data.
If the transformed data from taking the
log of just one of the variables is linear, this represents exponential growth.
If you get linear data from taking the log of both variables, this
is the power function.
I’ll be adding keys to the
problems from class here, so check back later.
More Transformation Problems Worksheet
More
Tranformations Problems Key
Problem
4.7 key
Problem
4.8 key
Problem
4.21 key
See me if you feel lost or
overwhelmed by these problems.
Mr P
Transformations
November 7, 2007
Transformations can be ugly, scary and monstrous at times. Don’t let them get you down. Make sure you know how to manipulate logs (or lns). This is important and makes the math less daunting.
The goal of transformations is to transform nonlinear data into a straight line. If you get straight line from taking the log on just one of the variables, this represents exponential growth. If you get a straight line from taking the log of both variables, this is the power function.
I’ll be adding keys to the problems from class here, so check back later.
See me if you feel lost or overwhelmed by these problems.
Mr P
1st Period - AP Stats
2nd Period - AP Stats
3rd Period - AA Finite
