Download Algebra 1 Summer Institute 2014 The ESP Verification Summary In

Algebra 1 Summer Institute 2014
The ESP Verification
Summary
Goals
In this activity,
participants will explore
some basic ideas about
probability, and some of
the relationships between
probability and statistics.
Participant Handouts
 Explore random events
 Explore games of chance
 Finite, equally likely
probability models
 Mathematical
probabilities and the
probability table
 Tree diagrams
 Pascal Triangle
 The binomial probability
model
1. The ESP Verification
Materials
Technology
Source
Estimated Time
Paper
Dice
Colored Pencils
LCD Projector
Facilitator Laptop
Excel
Annenberg Learner
website
120 minutes
Mathematics Standards
Common Core State Standards for Mathematics
MAFS. 7.SP.3: Investigate chance processes and develop, use, and evaluate probability
models
3.5: Understand that the probability of a chance event is a number between 0 and 1
that expresses the likelihood of the event occurring. Larger numbers indicate
greater likelihood. A probability near 0 indicates an unlikely event, a probability
around ½ indicates an event that is neither unlikely nor likely, and a probability
near 1 indicates a likely event.
3.6: Approximate the probability of a chance event by collecting data on the chance
process that produces it and observing its long-run relative frequency, and predict
the approximate relative frequency given the probability. For example, when
rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200
times, but probably not exactly 200 times.
3.7: Develop a probability model and use it to find probabilities of events. Compare
probabilities from a model to observed frequencies; if the agreement is not good,
explain possible sources of the discrepancy.
a. Develop a uniform probability model by assigning equal probability to all
outcomes, and use the model to determine probabilities of events. For
example, if a student is selected a random from a class, find the probability
that Jane will be selected and the probability that a girl will be selected.
b. Develop a probability model (which may not be uniform) by observing
1
Algebra 1 Summer Institute 2014
frequencies in data generated from a chance process. For example, find the
approximate probability that a spinning penny will land heads up or that a
tossed paper clip will land open end down. Do the outcomes for the spinning
penny appear to be equally likely based on the observed frequencies.
3.8: Find the probabilities of compound events using organized lists, tables, trees, and
simulation.
a. Understand that, just as with simple events, the probability of a compound
event is the fraction of outcomes in the sample space for which the compound
event occurs.
b. Represent sample spaces for the compound events using methods such as
organized lists, tables and tree diagrams. For an event described in everyday
language (e.g.,”rolling double sixes”), identify the outcomes in the sample
space which compose the event.
c. Design and use a simulation to generate frequencies for compound events. For
example, use random digits as a simulation tool to approximate the answer to
the question: If 40% of donors have type A blood, what is the probability that
it will take at least 4 donors to find one with type A blood?
MAFS.912.S-IC.1: Understand and evaluate random processes underlying statistical
experiments
1.1: Understand statistics as a process for making inferences about population
parameters based on a random sample from that population.
1.2: Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. For example, a model says that a
spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row
cause you to question the model?
Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use tools appropriately
Instructional Plan
In this unit, we will explore some basic ideas about probability, a subject that has
important applications to statistics.
1. Begin by asking participants to brainstorm in groups of 4 or 5 what is probability
and what does probability have to do with statistics. Ask them to write their
thoughts down on a poster paper and later have a class discussion reviewing their
ideas. What ideas do the groups have in common? (Slide 2)
2
Algebra 1 Summer Institute 2014
2. When many people think of probability, they think of rolling dice, picking
numbers at random, or playing the lottery. In fact, games of chance, which often
involve dice or other random devices, rely on the principles of probability. Ask
them what they think is a random event. Can they provide examples of something
that is random and something that is not?
3. In the next activity, we are going to test if the participants have ESP (extrasensory perception) while playing a game with one die. This activity is an
example of the binomial probability model, since there are two possible
outcomes: correct and incorrect. (Slide 3)
To play the game, they will need to be in groups of two. One player will roll the
die making sure the second player does not see the result. The second player,
using ESP, will predict if the die came up with an even or odd number. If the
second player made a correct prediction it counts as a correct call (C); otherwise it
is incorrect (I). Two rolls of the die count as one round of the game. Each player
should play for 20 rounds and record the results of each round in the following
table:
Player Toss 1 Toss 2 Player Toss 1 Toss 2
1
(C/I)
(C/I)
2
(C/I)
(C/I)
1
1
2
2
3
3
4
4
5
5
6
6
7
7
9
9
10
10
11
11
12
12
13
13
14
14
15
15
16
16
17
17
18
18
19
19
20
20
4. After they are done playing the game, they should complete a frequency table
counting how many time in each round they were correct. For example, in each
round they were either 0 times correct, 1 time correct, or 2 times correct. (Slide 4)
Number Correct Frequency Relative Frequency
3
Algebra 1 Summer Institute 2014
0
1
2
5. Pool the results from the whole class to complete a similar table.
6. Explain that they have collected experimental data.
We cannot know the outcome of a single random event in advance. However, if
we repeat the random experiment over and over and summarize the results, a
pattern of outcomes begins to emerge. We can determine this pattern by repeating
the experiment many, many times, or we can also use mathematical probabilities
to describe the pattern. In statistics, we use mathematical probabilities to predict
the expected frequencies of outcomes from repeated trials of random experiments.
Now we would like to come up with a mathematical model that could describe
someone without ESP and it is simply guessing. Participants should work in
groups of 4 and try to come up with a mathematical model. Ask a volunteer group
to share their results.
The possible outcomes for this investigation are: (Slide 5)
Roll 2
Roll 1
C
I
C
CC
CI
I
IC
II
From the table we have:
# Correct Number Fraction Decimal
0
1
1/4
.25
1
2
2/4
.50
2
1
1/4
.25
7. Compare the mathematical model results with their experimental results.
Hopefully they are very close and we can conclude that the participants were
guessing and do not have ESP. (Slide 6)
8. Taking the investigation further, ask participants what would happen if we roll the
die three times, or four times. How many do we get correct? Counting the number
correct is what is referred to as the binomial random variable.
4
Algebra 1 Summer Institute 2014
What outcomes could we get if we roll the die three times (n = 3)? We could get
0, 1, 2, or 3 corrects. If we roll the die 4 times, we can get: 0, 1, 2, 3, or 4 correct.
Ask participant to work in groups and come up with a mathematical model for the
cases when n = 3 and n = 4.
For example for n = 3, they should get: (Slide 7)
# Correct
0
1
2
3
Outcomes
Frequency Proportion
III
1
1/8
IIC, ICI, CII
3
3/8
ICC, CIC, CCI
3
3/8
CCC
1
1/8
For n=4 (Slide 8)
# Correct
0
1
2
3
4
Outcomes
Frequency Proportion
IIII
1
1/16
IIIC, IICI, ICII, CIII
4
4/16
IICC, ICIC, ICCI, CIIC, CICI, CCII
6
6/16
ICCC, CICC, CCIC, CCCI
4
4/16
CCCC
1
1/16
9. How could we figure out the number of total outcomes without listing them?
Tree diagrams can help with the counting. Demonstrate on the board how this
would look like
10. How many outcomes would there be if we roll the die 5 times? What is the
pattern? (Slide 10)
The pattern is powers of 2 where the exponent is equal to the number of times the
die is being rolled (2n).
11. The numbers we have seen in the frequency are 1, 3, 3, 1 and 1, 4, 6, 4, 1. Maybe
they have seen these numbers in the Pascal’s Triangle. Show the first 4 rows of
the triangle and explain how the numbers can be generated. (Slide 11)
5
Algebra 1 Summer Institute 2014
The rows of Pascal's triangle are conventionally enumerated starting with
row n = 0 at the top. The entries in each row are numbered from the left beginning
with k = 0 and are usually staggered relative to the numbers in the adjacent rows.
A simple construction of the triangle proceeds in the following manner. On row 0,
write only the number 1. Then, to construct the elements of following rows, add
the number above and to the left with the number above and to the right to find
the new value. If either the number to the right or left is not present, substitute a
zero in its place. For example, the first number in the first row is 0 + 1 = 1,
whereas the numbers 1 and 3 in the third row are added to produce the number 4
in the fourth row. The numbers in blue in the diagram represent the sum of the
entries in each row. For example, the sum of the entries in row 0 is 1, the sum of
the entries in row 1 is 2, and so on.
12. Pascal’s triangle provides the counts associated with each of the different number
correct we could get. How many outcomes could we get with 5 rolls? How many
of each count of corrects? (Slide 12)
Tree diagrams are useful to a certain point because they become large, however,
Pascal’s triangle give the numbers very easily.
13. How could we arrive to the mathematical probabilities for each of the numbers of
corrects that you might get if we did the experiment with 5 rolls of the die?
The mathematical probability is the number of counts divided by the total number
of outcomes.
The numbers in the next row of Pascal’s triangle would be: 1, 5, 10, 10, 5, 1, for a
total of 32. For 5 rolls of the die, the probabilities would be:
1/32 5/32 10/32 10/32 5/32 1/32
14. Ask participants to extend Pascal's Triangle to the 10th row. Using the 10th row,
determine the probability of tossing exactly five heads out of 10 coin tosses.
(Slide 13)
6
Algebra 1 Summer Institute 2014
The 10th row is 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1. The frequency of
five heads in 10 coin tosses is the sixth number in this row (remember that the
count starts with 0 heads), which is 252 (note that it is the center number in the
row). Since there are 210 = 1,024 possible outcomes in this row, the probability of
getting five heads out of 10 tosses is 252/1,024, or about 24.6%.
15. We've been investigating the binomial probability model. In a random experiment
with two possible outcomes, this model can be used to describe the probability of
either result. (Slide 14)
Consider, for example, a True-False test. If a test has four True-False questions,
and you make an independent guess on each question, how many will you get
correct? (Of course, the only thing you can say for sure is that you will get zero,
one, two, three, or four questions correct!)
Use the binomial probability model to determine the following, using the four
true-false questions example:
a. What is the most probable score you will get?
b. What is the least probable score you will get?
c. What is the probability of getting at least two answers correct?
d. What is the probability of getting at least three answers correct?
Answers:
a. The most probable score is two correct. It has a probability of 6/16.
b. The least probable scores are zero correct and four correct. Each has a
probability of 1/16.
c. The probability of getting at least two answers correct is 6/16 + 4/16 + 1/16 =
11/16.
d. The probability of getting at least three answers correct is 4/16 + 1/16 = 5/16.
16. Find the probability of getting at least two questions right on a 10-question TrueFalse test (where you must guess on each question). (Slide 15)
The simplest way to approach this problem is to find the probability of getting
less than two correct, then subtracting this from one. The probability of getting
less than two correct is 1/1,024 + 10/1,024 = 11/1024, so the alternate probability
is
1 - 11/1,024 = (1,024/1,024) - 11/1,024 = 1,013/1,024, or approximately 98.9%.
17. The table below shows the summary of scores of 100 rounds of another player's
attempt to show his ESP powers. Do these scores suggest that this player has
developed some serious ESP? (Slide 16)
7
Algebra 1 Summer Institute 2014
Number of Correct Experimental Frequency
0
2
1
14
2
29
3
34
4
21
Let's use a probability table to compare the experimental probability for this
player to the probabilities for a random player:
Number of Experimental Experimental Probability for
Correct
Frequency
Probability Random Player
0
2
.02
.0625
1
14
.14
.2500
2
29
.29
.3750
3
34
.34
.2500
4
21
.21
.0625
This player seems to have improved his ESP powers. In particular, this player's
experimental probability of getting four correct in four tries is more than three
times larger than the expected probability for a random player. This suggests that
this player has ESP (?).
8