Transcript Dealing With Uncertainty

Introduction to Simulations
and Experimental Probability
Introduction to Probability
Conditions for
a “fair game”

a game is fair if…


all players have an
equal chance of
winning, or
each player can
expect to win or lose
the same number of
times in the long run
http://mathdemos.gcsu.edu/plinko/
http://probability.ca/jeff/java/utday/
http://math.dartmouth.edu/~dlittle/java/Plinko/
Important vocabulary

a trial is one repetition of an experiment

random variable: a variable whose value
corresponds to the outcome of a random
event
Vocabulary / Terminology

expected value: (informally) the value to
which the average of the random variable’s
values tends after many repetitions; also
called the “mean”

event: a set of possible outcomes of an
experiment

simulation: an experiment that models an
actual event
A Definition of Probability

A measure of the likelihood of an event is
called the probability of the event.

It is based on how often a particular event
occurs in comparison with the total number of
trials.

Probabilities derived from experiments are
known as experimental probabilities.
Experimental Probability

Experimental probability is the observed
probability (also known as the relative
frequency) of an event, A, in an experiment.

It is found using the following formula:
P(A) = number of times A occurs
total number of trials
Note: probability is expressed as a number between 0 and 1
Exercises and Assignment

The Coffee Game is an example of a simulation (an
experiment that models an actual event).


In pairs, work through Investigation 1 (pp. 203-204)
Complete the 3 discussion questions on p. 204

Read through Example 2 -- solution 1, p. 207

Homework: p. 209, #2 (omit 2b i) and p. 211, #9
Theoretical Probability
Chapter 4.2 – An Introduction to Probability
Mathematics of Data Management (Nelson)
MDM 4U
Author: Gary Greer and James Gauthier
(with K. Myers)
Gerolamo Cardano

Born in 1501, Pavia,
Duchy of Milan
(today, part of Italy)

Died: 1571 in Rome

Physician, inventor,
mathematician,
chess player,
gambler
Games of Chance

Most historians agree that the modern study
of probability began with Gerolamo Cardano’s
analysis of “Games of Chance” in the 1500s.

http://encyclopedia.thefreedictionary.com
/Gerolamo Cardano

http://www-gap.dcs.st-and.ac.uk/~history
/Mathematicians/Cardan.html
A few terms…

simple event: an event that consists of
exactly one outcome

sample space: the collection of all possible
outcomes of the experiment

event space: the collection of all outcomes of
an experiment that correspond to a particular
event
General Definition of Probability

assuming that all outcomes are equally likely,
the probability of event A is:
P(A) = n(A)
n(S)

where n(A) is the number of elements in the
event space and n(S) is the number of
elements in the sample space.
An Example

When rolling a single die, what is the
probability of…

a) rolling a 2?
A = {2},
S = {1,2,3,4,5,6}
P(A) = n(A) = 1
n(S)
6
Example #1 (Part 2)

When rolling a single die, what is the
probability of…

b) rolling an even number?
A = {2,4,6},
S = {1,2,3,4,5,6}
P(A) = n(A) = 3 = 1
n(S)
6
2
Example #1 (Part 3)

When rolling a single die, what is the
probability of…

c) rolling a number less than 5?
A = {1,2,3,4},
S = {1,2,3,4,5,6}
P(A) = n(A) = 4 = 2
n(S)
6
3
Example #1 (Part 4)

When rolling a single die, what is the
probability of…

d) rolling a number greater than or equal to 5?
A = {5,6}, S = {1,2,3,4,5,6}
P(A) = n(A) = 2 = 1
n(S)
6
3
The Complement of a Set

The complement of a set A, written A’, consists
of all outcomes in the sample space that are not
in the set A.

If A is an event in a sample space, the
probability of the complementary event, A’,
is given by:
P(A’) = 1 – P(A)
Example #2

When selecting a single card from a complete
deck (no Jokers), what is the probability you
will pick…

a) the 7 of Diamonds?
P(A) = n(A) = 1
n(S)
52
Example #2 (Part 2)

When selecting a single card from a complete
deck (no Jokers), what is the probability you
will pick…

b) a Queen?
P(A) = n(A) = 4 = 1
n(S)
52 13
Example #2 (Part 3)

When selecting a single card from a complete
deck (no Jokers), what is the probability you
will pick…

c) a face card?
P(A) = n(A) = 12 = 3
n(S)
52 13
Example #2 (Part 4)

When selecting a single card from a complete
deck (no Jokers), what is the probability you
will pick…

d) a card that is not a face card?
P(A) = n(A) = 40 = 10
n(S)
52 13
Example #2 (Part 5)

Another way of looking at P(not a face card)…


we know: P(face card) = 3
13
and, we know: P(A’) = 1 - P(A)

So…
P(not a face card) = 1 - P(face card)
P(not a face card) = 1 - 3 = 10
13 13
Example #2 (Part 6)

When selecting a single card from a complete
deck (no Jokers), what is the probability you
will pick…

d) a red card?
P(A) = n(A) = 26 = 1
n(S)
52
2
Assignment, etc.

Read “Taking a Chance” by Rene Ritson:
http://www.infj.ulst.ac.uk/NI-Maths
/hypotenuse/volume13/Ritson.html
Next class: A look at Venn Diagrams
Probability and Chance
Probability




Probability is a measure of how likely it is for
an event to happen.
We name a probability with a number from 0
to 1.
If an event is certain to happen, then the
probability of the event is 1.
If an event is certain not to happen, then the
probability of the event is 0.
Probability

If it is uncertain whether or not an event will
happen, then its probability is some fraction
between 0 and 1 (or a fraction converted to a
decimal number).
B
C
A
D
3
1
2
A
C B
1. What is the probability that the spinner
will stop on part A?
2. What is the probability that the
spinner will stop on
(a) An even number?
(b) An odd number?
3. What fraction names the
probability that the spinner will
stop in the area marked A?
Probability Activity




In your group, open your M&M bag and put
the candy on the paper plate.
Put ten brown M&Ms and five yellow M&Ms
in the bag.
Ask your group, what is the probability of
getting a brown M&M?
Ask your group, what is the probability of
getting a yellow M&M?
Examples



Another person in the group will then put in 8
green M&Ms and 2 blue M&Ms.
Ask the group to predict which color you are
more likely to pull out, least likely, unlikely, or
equally likely to pull out.
The last person in the group will make up
his/her own problem with the M&Ms.
Probability Questions

Lawrence is the captain of his track team.
The team is deciding on a color and all eight
members wrote their choice down on equal
size cards. If Lawrence picks one card at
random, what is the probability that he will
pick blue?
blue
blue
yellow
red
green
black
blue
black

A.
B.
C.
Donald is rolling a number cube labeled 1 to
6. Which of the following is LEAST
LIKELY?
an even number
an odd number
a number greater than 5
CHANCE

Chance is how likely it is that something will
happen. To state a chance, we use a
percent.
0
½
1
Probability
Certain not
to happen
Equally likely to
happen or not to happen
Certain to
happen
Chance
0%
50 %
100%
Chance

When a meteorologist states that the chance
of rain is 50%, the meteorologist is saying
that it is equally likely to rain or not to rain. If
the chance of rain rises to 80%, it is more
likely to rain. If the chance drops to 20%,
then it may rain, but it probably will not rain.
1 2
4 3
1. What is the chance of spinning a
number greater than 1?
4 1
2
3
5
2. What is the chance of spinning a
4?
3. What is the chance that the
spinner will stop on an odd
number?
4. What is the chance of rolling an even
number with one toss of on number cube?
DRILL
1)
2)
3)
What is the probability of rolling an odd
number on a 6-sided die?
What is the probability of getting a green
marble, if there is a bag with 6 blue
marbles, 5 green marbles, 3 yellow
marbles and 8 orange marbles?
What is the probability of not getting a
blue or yellow marble from the same
bag?
Tree Diagram

Is a method used
for writing out all
the possible
outcomes for
multiple events.
Sample Space


The sample space is the set of all
possible outcomes for a given event.
Example: The sample space for rolling a
die is {1, 2, 3, 4, 5, 6}
Counting Principle



If two or more events occur in x and y
ways to find the total number of
combinations (choices) you simply multiply
the number of possible outcomes in each
group by each other.
Example: If you have 4 shirts, 3 pairs of
pants, 2 pairs of shoes and 3 hats, you
can make 4(3)(2)(3) different outfits.
Which gives you a total of 72 outfits.
Factorial




Is used when you want to figure out how
many ways “n” number of objects can be
arranged.
The symbol for factorial is an exclamation
point. (n!)
Factorial means to multiply by every
number less then “n” down to 1.
Example: 5! = 5(4)(3)(2)(1)
DRILL
1)
2)
3)
What is the probability of rolling a number
less than 5 on a 6-sided die?
What is the probability of getting a green
marble, if there is a bag with 4 blue
marbles, 3 green marbles, 4 yellow
marbles and 9 orange marbles?
What is the probability of not getting a
green or yellow marble from the same
bag?
Classwork
Pages 756 – 757
#’s 1 – 22, 25, 26
DRILL
1)
2)
3)
What is the probability of rolling a number
less than 5 on a 6-sided die?
What is the theoretical probability of
getting a green marble, if there is a bag
with 16 blue marbles, 18 green marbles,
14 yellow marbles and 16 orange
marbles?
What is the experimental probability of
rolling a 3 given:
{2, 3, 4, 1, 3, 4, 6, 5, 2, 3, 3, 4, 1, 6}
Algebra I
Experimental vs. Theoretical
Probability
Theoretical Probability

The theoretical probability of an event is
the “actual” probability of something
happening.

Number of “correct” outcomes divided
by the total number of outcomes.
Experimental Probability

The experimental probability of an event is
the probability of an event based on
previous outcomes.

Example: If you flipped a coin 10 times
and got { T, T, H, T, H, H, H, T, T, T}

The experimental probability of getting
tails is 6 out of 10 or 3/5.
Example
a)
{2, 4, 1, 6, 5, 1, 1, 4, 5, 3, 2, 3, 6, 6}
b)
{1, 4, 5, 5, 3, 3, 6, 2, 6, 6, 2, 3, 4, 1}
c)
{2, 3, 2, 2, 5, 6, 1, 2, 3, 4, 3, 2, 2, 6}
Calculator Activity
* We are going to simulate rolling a die 50
times using the calculators and then
calculate the theoretical probability and
experimental probability of the event.
Homework

Write five events and say what the
theoretical probability and experimental
probability of the events are.
Ex: {1, 4, 3, 5, 5, 2, 3, 1, 6, 2, 2}

Theoretical Prob of rolling a 2 is 1/6

Experimental Prob of rolling a 2 is 3/11

Stick with Probability
Objectives:



Students will learn new vocabulary terms
relating to probability.
Students will determine the probability of
different events.
Students will develop a pie chart for their
probability data.
Vocabulary

Probability

Certain

Impossible

Likely

Unlikely

Possible Outcomes

Experimental Probability
Vocabulary

Probability: chance that an
event will happen (study of
chance)

Unlikely: greater chance
that it will not happen, than
it will happen

Certain: that event will
happen

Possible Outcomes:
number of outcomes you
are able to achieve

Impossible: that event will
never happen

Experimental Probability:
can be found by conducting
repeated trials

Likely: greater chance that it
will happen, than not
happen
Probability of an event….
Probability of an event =
number of favorable outcomes
number of possible outcomes
Demonstration….
What would be the probability of
picking a green marble?

Green Marble =
5
8
If we have a bag of colored sticks?




1. What is the probability in and getting a
yellow?
2. Green?
3. Red?
4. Blue?
More Questions???



5. How could you change the sticks in the
bag so that the chances are just as good at
getting one color as it is another?
6. What color has the greatest chance of
being picked?
7. Least chance of being picked?
Probability Activity…. Have Fun!




4-5 Groups
bag of colored sticks (don’t peak!)
*two yellow
*seven blue
*three red
*five green
tally chart
dry-erase markers (only on the chart!)
Directions….





You are going to (as a group) draw out of the bag 15
times, taking turns.
Each time you draw a stick, draw a tally under the
color you drew on the chart.
Afterwards, add each color up to get the total. Write
these totals under each color in the indicated box.
TOMORROW… we will work on the computer and
make a pie graph that represents our data.
GOOD LUCK!
What are some other probability tools?






Dice (number cubes)
Colored Candies
Colored Pipe Cleaners
Marbles
Spinner
Coin Toss
Resources

http://images.google.com/images?q=Probabil
ity
9-3 Use a Simulation
Warm Up
Problem of the Day
Lesson Presentation
Pre-Algebra
9-3 Use a Simulation
Warm Up
1. There are 25 out of 216 sophomores enrolled in
a physical-education course. Estimate the
probability that a randomly selected sophomore
is enrolled in a physical-education course.
0.12
2. A spinner was spun 230 times. It landed on red
120 times, green 65 times, and yellow 45 times.
Estimate the probability of its landing on red.
0.52
Pre-Algebra
Problem of the Day
If a triangle is worth 7 and a rectangle is
worth 8, how much is a hexagon worth?
10
Learn to use a simulation to estimate
probability.
Vocabulary
simulation
random numbers
A simulation is a model of a real
situation. In a set of random numbers,
each number has the same probability of
occurring as every other number, and no
pattern can be used to predict the next
number. Random numbers can be used to
simulate random events in real situations.
Additional Example 1: Problem Solving Application
A dart player hits the bull’s-eye 25%
of the times that he throws a dart.
Estimate the probability that he will
make at least 2 bull’s-eyes out of his
next 5 throws.
1
Understand the Problem
The answer will be the probability that he
will make at least 2 bull’s-eyes out of his next
5 throws.
List the important information:
• The probability that the player will hit the
bull’s-eye is 0.25.
Additional Example 1 Continued
2
Make a Plan
Use a simulation to model the situation.
Use digits grouped in pairs. The numbers 01–
25 represent a bull’s-eye, and the numbers
26–00 represent an unsuccessful attempt.
Each group of 10 digits represent one trial.
Additional Example 1 Continued
2
Make a Plan
Additional Example 1 Continued
3
Solve
Starting on the
third row of the
table from the
previous slide and
using 10 digits for
each trial yields
the data at right:
Additional Example 1 Continued
Out of the 10 trials, 2 trials represented two or
more bull’s-eyes. Based on this simulation, the
probability of making at least 2 bull’s-eyes out
of his next 5 throws is about 2 , or 20%.
10
4
Look Back
Hitting the bull’s-eye at a rate of 20% means
the player hits about 20 bull’s-eye out of
every 100 throws. This ratio is equivalent to
2 out of 10 throws, so he should make at
least 2 bull’s-eyes most of the time. The
answer is reasonable.
Try This: Example 1
Tuan wins a toy from the toy grab machine
at the arcade 30% of the time. Estimate the
probability that he will win a toy 1 time out
of the next 3 times he plays.
1
Understand the Problem
The answer will be the probability that he
will win 1 of the next 3 times.
List the important information:
• The probability that Tuan will win is 30%.
Try This: Example 1 Continued
2
Make a Plan
Use a simulation to model the situation. Use
digits grouped in pairs. The numbers 01–30
represent a win, and the numbers 31–00
represent an unsuccessful attempt. Each
group of 6 digits represent one trial.
Try This: Example 1 Continued
3
Solve
Starting on the
fourth row of
the table from
slide 3 and
using 6 digits
for each trial
yields the data
at right:
86 58 52
79 19 65 1 win
26 49 35 1 win
57 94 42
51 33 25
16 63 52
1 win
85 84 18 1 win
39 47 32
66 67 89
93 87 83
Try This: Example 1 Continued
Out of the 10 trials, 4 trials represented one
or more wins. Based on this simulation, the
probability of winning at least 1 time out of
his next 3 games is 40%
4
Look Back
Winning at a rate of 40% means that Tuan
wins about 40 times out of every 100 games.
This ratio is equivalent to 4 out of 10 games,
so he should win at least 4 toys most of the
time. The answer is reasonable.
Lesson Quiz
Use the table of random numbers to simulate
the situation.
38094
76211
43659
29272
76005
93391
19587
47380
33442
40809
27904
95412
69632
48461
25654
55889
69632
48461
25654
55889
42231
39983
13802
24483
52730
15604
80949
46351
10580
59765
76431
38586
62987
40440
93594
30198
64926
17672
68735
35168
19085
35497
30798
21966
Lydia gets a hit 34% of
the time she bats.
Estimate the probability
that she will get at least
4 hits in her next 10 at
bats.
Possible answer: 30%
Examples
1)
2)
3)
A bag contains 30 marbles 12 red, 8 blue, 4
orange and 6 green. What is the probability
that you do not pick a blue.
If you flip two coins 40 times and you got
HH 11 times, HT 9 times, TH 12 times and
TT 8 times. What is the experimental
probability of getting HH?
What is the probability of getting a vowel
from the word “MATHEMATICS”
Chapter 6 Review
Probability
TEST TOMMOROW
Examples
4) How could you do a simulation for making a
shot in basketball if you had a 60% chance
of making the shot and you were shooting 5
times?
5) If 12 out of 50 students took carpentry how
many would you project take carpentry out of
2000 students?
6) If 22 out of 80 students go on vacation over
the summer, how many would you expect to
go on vacation if there were 480 students?
Examples
7) How many different outfits could you make if
you had 5 pairs of pants, 6 shirts and 4 pairs
of shoes?
8) If a given teacher has 8 freshman, 14
sophomores, 16 juniors and 22 seniors, what
is the probability that the teacher will randomly
select one student and it will be a sophomore?
9) What tools could you use to simulate a 40%
probability?
Examples
10)How many students out of 3,500 would
you expect watch over 4 hours of T.V a
week if 24 out of 125 students watch
over 4 hours of T.V a week?
11) Draw a spinner that might have been
used that if it was spun 100 times and it
landed on the number one 25 times, the
number two 50 times the number three 12
times and the number four 13 times?
Examples
ECR