Download Day 1 - MSTE

Goal 1: To help practicing teachers get a fundamental
understanding of t-tests and p-value in a short amount of time
and apply it to biology data they obtain, without “too much
math.”
2-sample t-tests
p-value approach
Sampling distributions & CLT
Continuous probability distribution
Probability Fundamentals
Goal 2: Learn to use TI-84 Graphing Calculator to do statistical
analysis and find p-values and understand what the results mean.
AIMS – statistics workshop Day 1 page 1
Goal for Day 1:
Develop an intuitive understanding of Probability without giving a
“lesson” on rules of probability and don’t make it seem like math;
Walk through some ideas/activities for teaching probability;
Look at probability items on the ISAT;
Look at probability in Connected Math;
Develop an understanding of probability distributions.
AIMS – statistics workshop Day 1 page 2
YOU DECIDE: Is Probability involved in the solution?
1. You buy two identical notebooks at different stores.
One notebook cost $1.89, the other $.99. You started
out with $11.00. How much money do you have left?
2. You set our VCR to record a TV show at 7:30 p.m. on
Channel 2. You come home at 9:30 pm. Do you know
what is recorded on the tape?
3. Your new computer has just arrived. You open the box,
set up the computer, and flip the switch to "on." Will
it work?
Moral: If outcomes are uncertain -> it's probability
AIMS – statistics workshop Day 1 page 3
Probability theory had its origin in the 16th Century, when
an Italian physician and mathematician named Jerome
Cardan wrote the first book on the subject, The Book on
Games of Chance. For many years the "mathematics of
chance" was used primarily to solve gambling problems.
It has come a long way since then. Today, the theory of
probability is, according to some mathematicians, a
"cornerstone of all the sciences." People use probability
to predict sales, plan political campaigns, determine
insurance premiums and much more!
Founders: Blaise Pascal (1623-1662)
Pierre de Fermat (1601-1665)
AIMS – statistics workshop Day 1 page 4
Classic probability problems:
Birthday Paradox
Monty's Dilema
Cereal Box Problem
Small World Problem
Real-world applications of probability:
Relative Risk
Search and Rescue
Queuing Theory
Weather Forecasting
Car and Life Insurance
Lotteries and Gambling
AIMS – statistics workshop Day 1 page 5
Probabilities are RATIOS, expressed as fractions
decimals, or percents, determined by considering results
or outcomes of experiments.
Experiment = an activity whose results are determined by chance
such as tossing a coin, rolling dice, drawing marbles from a bag
Outcome = a result of an experiment
such as tossing a head, rolling a 2, or drawing a red marble
Sample Space = the list (set) of all outcomes
such as H or T, {1,2,3,4,5,6}, or {Red, Blue, White} marbles
Event = a subset of a sample space
AIMS – statistics workshop Day 1 page 6
Probability
Classical Rule or Theoretical Rule:
# of times A occurs in sample space
P(A) =
total # of events in sample space
A
B
1
2
3
4
5
6
1
2
3
4
5
6
C
1
2
3
4
5
6
AIMS – statistics workshop Day 1 page 7
Empirical Rule or Experimental Rule:
# of times A is observed
P(A) =
total # of trials
A
B
1
2
3
4
5
6
1
2
3
4
5
6
C
1
2
3
4
5
6
AIMS – statistics workshop Day 1 page 8
Subjective Rule:
# of times A is believed to occur
P(A) =
total # of trials
B
A
C
1
2
3
4
5
6
AIMS – statistics workshop Day 1 page 9
Five Great Activities Using a Spinner
In the circle, which cell will the spinner most likely land
on most of the time?
A
1
2
3
4
5
6
C
1
2
3
4
5
6
B
1
2
3
4
5
6
D
1
2
3
4
5
6
AIMS – statistics workshop Day 1 page 10
For this experiment, you will need the spinners below. First, have
one team member to use spinner P, another to use spinner R, and the
third to record the results. The two players simultaneously flick
their spinners and the spinner landing on the higher number wins.
Each pair of players should do 25 trials. After pooling the results
from the class, which spinner wins most of the time?
Spinner P
Spinner R
20
60
70
10
50
90
AIMS – statistics workshop Day 1 page 11
Spinner S
30
40
80
Next, have the third member take spinner S, and play against the
second player using spinner R. Spin 25 times. After pooling the
results from the class, which spinner wins most of the time?
Finally, have the first member use spinner P play against the third
member using spinner S. Spin 25 times. After pooling the results
from the class, which spinner wins most of the time?
AIMS – statistics workshop Day 1 page 12
Mathematical analysis: Calculate the probability of P beating R by
making a table of equally likely outcomes.
Spinner R
10 50 90
20 P R R
Spinner P
60 P P R
70 P P R
Make similar tables for R and S, and for S and P and summarize the
findings.
Spinner S
Spinner R
Spinner P
Spinner S
AIMS – statistics workshop Day 1 page 13
In Algebra, there is a transitive property.
Ex: 4/12 = 2/6 = 1/3, then 4/12 = 1/3
In Logic, there is a reasoning argument called the chain rule (also
known as hypothetical syllogism).
ex: If I save, I will retire early.
If I retire early, I will become lazy.
Therefore, If I save, I will become lazy.
In general, "if p then q" and "if q then r" are true,
then "if p then r" is true.
Do the spinners illustrate the transitive property or the chain rule?
Do you have any idea why the spinners are named P, R, and S?
AIMS – statistics workshop Day 1 page 14
From the ISAT Math Materials, grade 3
1.
There are three red, two blue and two green erasers in a bag.
What are the chances of picking a green eraser out of the bag
without looking?
a. one out of seven chances
b. two out of seven chances
c. three out of seven chances
d. four out of seven chances
2. Amber has a box of 7 red, 22 green, 10 yellow and 2 blue paper
clips. If she picked one out of the box, what color will it most
likely be?
a. blue
b. green
c. red
d. yellow
AIMS – statistics workshop Day 1 page 15
From the ISAT Math Materials, grade 5
1.
Ellis has a box of 8 art pencils: 1 red, 3 blue, and 4 yellow. What
is the probability of pulling a red pencil from the box?
a. 1/8
b. 1/3
c. 3/8
d. 8/3
e. 3/1
2. Benjamin has five playing cards. They are:
Black 3
Red 6
Red
4
Red 7
Black 6
If a card is drawn without looking, what is the probability that it
will be an even numbered red card?
a. 2/3
b. 1/5
c. 2/5
d. 3/5
e. 4/5
AIMS – statistics workshop Day 1 page 16
From the ISAT Math Materials, grade 5
3. A table shows the result of 100 spins. The spinner has 6 equal
sections. What did the spinner probably look like?
Color
Number of times this color was spun
Orange
32
White
16
Gray
36
Tan
18
AIMS – statistics workshop Day 1 page 17
The Pesky Magician
A pesky Magician comes to the fair each year. He has card tricks and
dice games and enjoys surprising people so that they are in awe of
his powers. He seems only slightly sinister . . . almost likable . . . and
it is unnerving that he uses dice and cards and things you thought
you understood, but he always seems to win! You want to know more.
You think there must be some trick to what he is doing, because in all
your years, you’ve never seen someone repeatedly win at a game,
unless there was a trick or it was “rigged.” You watch him carefully . .
. you’re planning to challenge him soon! You’ve thought up a good
game, so you approach the Magician and say:
"I have a new game I would like to propose to you. Let's roll 2 dice
and ADD them. If the sum is even, you win. If the sum is odd, I win.
Your present him with a list of the possible sums:
2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12
AIMS – statistics workshop Day 1 page 18
But now that you see the list of possibilities, you panic! "Oh no!
What did I get myself into?" you ask. Of course, the Magician likes
what he sees because he seems to have the advantage. He thinks . . .
hmmmm . . . 6 are even and only 5 are odd. It looks like an even
answer has the advantage . . . true?
A way to look at the sample space when rolling 2 dice:
1
2
3
4
5
6
7
8
9
10
11
12
AIMS – statistics workshop Day 1 page 19
The pesky Magician proposes another game to play. He says: "Here
is a new game you'll like. You will roll two dice. If we subtract the
smaller amount from the larger amount, the possible answers are:
0-1-2-3-4-5
I'll take the answers 0 - 1 - 2 and you take 4 - 5 - 6. OK?
We tally the results. After 10 rolls, we see who wins.
The Magician likes to suggest games that are in his favor. You tell
him that you are not interested in playing that game, because it is
strongly in his favor. "Oh, but I really want to play a game with you,"
he says. "How about if I can show you a diagram for a game so you
can see all the possible combinations? If I can do that, will you
play?"
AIMS – statistics workshop Day 1 page 20
So the Magician explains, "In this game, you roll two dice and
multiply them. If the answer is an even number, I get a point. If the
answer is an odd number, you get a point. And I'm even going to show
you a diagram of the combinations . . ."
Possible combinations for the products of 2 dice:
1
1
1
1
1
1
2
2
2
2
2
2
3x1 3x2 3x3 3x4 3x5 3x6
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
AIMS – statistics workshop Day 1 page 21
Whoa! . . . hold the phone! . . . the pesky Magician watched me
diagram the possible outcomes. He saw my face when I discovered
the game is NOT fair, so he is very surprised when I look him in the
eye and say: "OK, Mr. Magician, I agree to play this game on one
condition . . . "
He is so startled to hear me say I will play that he says, "you will? I
mean, oh, good. What is the condition?"
"Every time there is an even number answer, you get one point.
Every time there is an odd number answer, I get three points. This
will adjust the game to make it fair. We will play until the first one
scores 21 points. Agree?"
AIMS – statistics workshop Day 1 page 22
To make an informed decision to play,
1. review the rules
2. picture the possibilities
3. regard the risks, and
4. state your chance of success and chances of losing. This will help
you decide what you are willing to risk when playing a game.
A game is FAIR only when it has equally likes outcomes of winning
and loosing.
Is Roulette FAIR?
AIMS – statistics workshop Day 1 page 23
Multi-Stage Events & Tree Diagrams
The probability of the outcome along any path is equal to
the product of all probabilities along the path.
Ex: Toss 2 coins
1st toss
2nd toss
H
H
T
H
T
T
AIMS – statistics workshop Day 1 page 24
Multi-Stage Events & Tree Diagrams
Ex: Roll 2 dice
1st die
1
2nd die
1
2
3
4
5
6
2
3
4
5
6
AIMS – statistics workshop Day 1 page 25
Multi-Stage Events & Tree Diagrams
The probability of the outcome along any path is equal to
the product of all probabilities along the path.
Socks in the Dark: You have a drawer filled with single socks.
If they were mated there would be 2 pair of white socks, 3
pair of tan socks, and 5 pair of black socks. Draw a tree
diagram that shows all of the possible ways of picking two
socks from the drawer in the dark one sock at a time, without
replacement. Then determine the probability that you pick a
pair of:
a) black socks
b) tan socks
c) the same colored socks
AIMS – statistics workshop Day 1 page 26
Tree Diagram for Socks-in-the-dark problem:
1st sock
2nd sock
Black
Black
Tan
White
Tan
Black
Tan
White
White
Black
Tan
White
AIMS – statistics workshop Day 1 page 27
Most probability problems in the ISAT can be solved using tree
diagrams, and without knowing rules of probability!
Suppose a box contains 4 red and 5 white marbles. Two marbles are
drawn. What is the probability they are both red? (note: you can
draw any number of marbles; you can find any combination of colors; you can
replace or not replace after each draw)
1st marble
2nd marble
Red
Red
White
Red
White
White
AIMS – statistics workshop Day 1 page 28
Connected Math: Connections page 37
A bag contains several marbles. Some are red, some are white, and
some are blue. Carlos counted the marbles and found that the
theoretical probability of drawing a red marble is 1/6 and the
theoretical probability of drawing a white marble is 1/3.
1. What is the smallest number of marbles that could be n the bag?
2. Could the bag contain 48 marbles? If so, how many of each color
must it contain?
3. If the bag contains 4 red marbles and 8 white marbles, how many
blue marbles must it contain?
4. How can you tell what the probability of drawing a blue marble is?
AIMS – statistics workshop Day 1 page 29
Connected Math: Unit Reflections page 65
Joanna designed a game for the school carnival. She prepared two
bags of marbles.
Bag A contains 3 marbles: 1 red, 1 blue, 1 green.
Bag B contains 4 marbles: 2 reds and 2 blues.
To play the game, a contestant picks one marble form each bag. If
the colors of the marbles match, the contestant wins a prize.
These are the win/loss results for the first 30 games.
W L L W W L L W W L L W W L L
L W L L L L W L W L L L L W W
a. What do these data suggest about the experimental probability of
winning the game?
b. What is the theoretical probability of winning the game?
c. What explains the difference between your answers to question a
and question b?
AIMS – statistics workshop Day 1 page 30
A Probability Distribution is a list of the outcomes from
an experiment along with their respective probabilities.
Experiment: toss 2 coins
Probability Distribution:
Heads
0
1
2
P(H)
1/4
1/2
1/4
The only rules for a probability distribution is that each
probability must be a number between 0 and 1, and the
sum of all the probabilities must add to 1 (or 100%).
AIMS – statistics workshop Day 1 page 31
Ex: Is the following a Probability Distribution?
When four different households are surveyed on Monday
night, the number of households with television tuned to
Monday Night Football on ABC with their relative
frequency is shown (based on data from Nielsen Media
Research).
MNF P(MNF)
0
0.522
1
0.368
2
0.098
3
0.011
4
0.001
AIMS – statistics workshop Day 1 page 32
Experiment: roll 2 dice
Probability Distribution:
sum
P(sum)
1
2
3
4
5
6
7
8
9
10
11
12
0
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36
AIMS – statistics workshop Day 1 page 33
13
0
Suppose a couple wants 4 children. Which is more likely,
3 of one sex and 1 of another, or 2 and 2 of each sex?
Sample Space for 4 children (2x2x2x2 = 16 outcomes)
GGGG
BGGG
What does the tree diagram look like?
GGGB
BGGB
GGBG
BGBG
GGBB
BGBB
GBGG
BBGG
GBGB
BBGB
GBBG
BBBG
GBBB
BBBB
AIMS – statistics workshop Day 1 page 34
Make a probability distribution for the number of girls.
Experiment: have 4 kids
Probability Distribution:
# girls
Prob(# girls)
0
1
2
3
4
5
1/16
3/16
8/16
3/16
1/16
0
AIMS – statistics workshop Day 1 page 35
Probability distributions may result from discrete or
continuous data. Tomorrow, we will look at a discrete
probability distribution, the binomial distribution. Then
we will look at a continuous probability distribution, the
normal distribution.
Probability Distributions
Discrete
* Binomial
Geometric
Continuous
* Normal
Chi-Square
AIMS – statistics workshop Day 1 page 36
Poisson
AIMS – statistics workshop Day 1 page 37