Goal 1: To help practicing teachers get a fundamental ...



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.”

Goal 2: Learn to use TI-84 Graphing Calculator to do statistical analysis and find p-values and understand what the results mean.

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.

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

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)

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

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

Probability

Classical Rule or Theoretical Rule:

P(A) = [pic]

Empirical Rule or Experimental Rule:

P(A) = [pic]

Subjective Rule:

P(A) = [pic]

Five Great Activities Using a Spinner

In the circle, which cell will the spinner most likely land on most of the time?

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

Spinner S

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?

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 |

|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 P

| | |

|Orange |32 |

|White |16 |

|Gray |36 |

|Tan |18 |

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

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:

| | | |

| |Heads |P(H) |

|Probability Distribution: |0 |1/4 |

| |1 |1/2 |

| |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%).

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 |

|Experiment: roll 2 dice |sum |P(sum) |

|Probability Distribution: |1 |0 |

| |2 |1/36 |

| |3 |2/36 |

| |4 |3/36 |

| |5 |4/36 |

| |6 |5/36 |

| |7 |6/36 |

| |8 |5/36 |

| |9 |4/36 |

| |10 |3/36 |

| |11 |2/36 |

| |12 |1/36 |

| |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

Make a probability distribution for the number of girls.

|Experiment: have 4 kids |# girls |Prob(# girls) |

|Probability Distribution: |0 |1/16 |

| |1 |3/16 |

| |2 |8/16 |

| |3 |3/16 |

| |4 |1/16 |

| |5 |0 |

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 Continuous

* Binomial * Normal

Geometric Chi-Square

Poisson

-----------------------

2-sample t-tests

p-value approach

Sampling distributions & CLT

Continuous probability distribution

Probability Fundamentals

B[pic]

A[pic]

C[pic]

C[pic]

B[pic]

A[pic]

C[pic]

B

A[pic]

A[pic]

B[pic]

D[pic]

C[pic]

Spinner P

1

2

3

4

5

6

Spinner S

70

20

60

10

50

90

30

40

80

Spinner R

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