Download 6Lessons7.3,7.4

Agenda
• Probability and Chance
– Event, outcome, equally likely, not equally likely
– Calculate probablity of an event
– Tree diagrams
Probability
• Write a definition.
• The likelihood of an event.
• The ratio of desired outcomes to total
outcomes.
• What should happen in an experiment.
• A number between 0 and 1.
• A percent between 0% and 100%.
Basic Concepts
• We write probabilities as ratios--these ratios
can then be written as fractions or percents.
• 0 means that the probability of something
happening is impossible.
• 1 means that the probability of something
happening is certain.
• Odds represent the ratio:
outcomes we want:outcome we don’t want.
More basic concepts
• The experiment
e.g., tossing a coin, picking 4 cards, weather
conditions, etc.
• Outcome: What could happen in the
experiment
e.g., getting a head or a tail, JJQ2 or A357
(or 6,497,400 others), rain, snow, sleet,
clouds, sun, etc.
• Event: What we want in an experiment
e.g., getting a head, picking all hearts, no
precipitation.
Still more basic concepts
• Equally likely: each outcome is as likely as
any other outcome
e.g., flipping a coin: H or T
• Not equally likely: some outcomes are more
likely than others.
e.g., winning or losing the lottery
• Random: cannot make an individual
prediction.
Let’s try a few
•
•
•
•
•
•
In a deck of 52 cards, in a single draw
P(red)
P(3)
P(3 or 5)
P(red 3)
P(Ace of clubs)
Let’s try a few
• In the spinner
at the right:
• P(yellow)
• P(not blue)
• P(red or yellow)
• P(white)
• P(odd number)
Spinner
Let’s try a few
• Roll 2 dice,
find the sum.
• P(7)
• P(11)
• P(even)
• P(odd)
• P(prime)
• P(1)
1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6
3,1 3,2 3,3 3,4 3,5 3,6
4,1 4,2 4,3 4,4 4,5 4,6
5,1 5,2 5,3 5,4 5,5 5,6
6,1 6,2 6,3 6,4 6,5 6,6
From the previous
examples…
•
•
•
•
•
•
•
•
Random
Equally likely
Not equally likely
Certain
Impossible
Disjoint events with no common outcomes
Mutually exclusive P(A) + P(B) = P(A or B)
Complement P(A) + P(B) = 1
Tree Diagrams
• If I give an “Always, Sometimes, Never”
test with 2 questions…
• P(2 always answers)
• Possible outcome: AA, NS, NA, etc.
• How many outcomes?
Tree Diagram
• If “no replacement” (duplicates allowed)
A
S
N
A S N
A S N
A S N
Each outcome has probability 1/9: 9 different
outcomes possible, only 1 with AA.
Spinner
Tree diagram
• If I spin the spinner at the right,
and then flip a coin:
R
H
B
T
H
G
T
H
W
T
H
T
• Let’s list some possible outcomes: RH, RT,
YT, BH, etc.
• How many outcomes are there? In this case,
there are 8: each outcome can be found by
a path on the tree diagram.
Spinner
Bigger Tree Diagram
• If I spin the spinner at the right,
and then flip a coin 2 times:
R
H
B
T
H T H T
G
H
T
H T
H T
H
H
W
T
T H
T
H
T
H T
H
T
Now there are 16 outcomes, such as RHH, GTT.
Tree Diagram
Spinner
• What if not all outcomes
are equally likely?
• If we spin this spinner
2 times.
1/4
1/2 B
B R Y
1/2 1/4 1/4
B
R
R Y
1/2 1/4 1/4
1/4 Y
B
R
Y
1/2 1/4 1/4
Think: weighted averages! P(B,Y) = .5 • .25
Exploration 7.19
• Step 1: Pick any six games--use all
categories except for Marshmallows.
• For each game, predict whether you think it
is fair or unfair. Write a 1-2 sentence
explanation.
• Then, play each game a few times.
• You have 10 - 15 minutes to complete this.
Exploration 7.19
•
•
•
•
•
•
Step 2: reread all of the games.
Write down the experiment,
the different outcomes that are possible, and
the outcomes desired for each event/player.
Do this in a systematic way.
Then, write whether the game is fair or not-for the six games you played, record whether
your prediction was correct or incorrect.
• 10 - 15 minutes for this part.
Theoretical vs.
Experimental
• In Exploration 7.19, even if a game is
unfair, both players may still win a
portion of all the games played. It is
even possible that the player who
“should” win will actually lose more
often than the person who “should”
lose. Why does this happen?
Theoretical vs.
Experimental
• Theoretical probability: What “should”
happen when you do the experiment.
• Experimental probability: What “did”
happen when you conducted the
experiment.
• Experimental probability is sometimes
called relative frequency of an event.
Theoretical vs.
Experimental
•
•
•
•
•
Experiment: flip a coin 2 times.
P(2 heads)
Possible outcomes: HH, HT, TH, TT
Theoretical Probability:
Experimental Probability:
Sample Size
• If we did the coin experiment 1 time, the
experimental probability will either be 1 or 0.
Either we got two heads (1) or we got two
tails (0) or we got one head and one tail (0).
So, in a sample of 1 trial, experimental ≠
theoretical.
• But, if we did the coin experiment 5 million
times, experimental ~ theoretical.
• This is called the Law of Large Numbers.
Warm Up
• Suppose I have 6 marbles--two are red
and 4 are white.
• If I can pick 2 marbles, what is the
probability that at least 1 is red?
• Hint: the answer is not 1/3 or 1/4.
• Possible outcomes: RR, RW, WR, WW
• However, If we make a tree diagram,
we can see this more clearly.
For the first pick, we will have 6 choices. After the
first pick, there will only be 5 marbles remaining.
• Count the total
outcomes.
How many have
no reds?
Subtract this
from total to
get P(≥ 1 red)
Shortcut!
P(1 red) + P(2 reds) = 4/6 • 2/5 + 2/6 • 4/5 + 2/6 • 1/5
4/6
3/5
2/6
2/5
4/5
1/5
Agenda
•
•
•
•
Go over warm up
Independent vs. dependent events
Exploration 7.21
Expected value
Why do we multiply for this
problem?
• Let’s look at it this way. When we
made the tree diagram, how many
outcomes were in the first “draw”? How
many outcomes were in the second
“draw”? How many outcomes are
shown in the tree diagram?
Try this one.
• Answer these questions just by looking at the
tree diagram.
Suppose I want to know the probability of
picking
1 white?
• 2 whites?
• Now, redo the problems using multiplication.
Independent vs.
Dependent
• With random events, when the second
outcome is not related to the first
outcome, we say they are independent.
• With random events, when the second
outcome is related to the first outcome,
we say they are dependent.
Independent vs.
Dependent
•
•
•
•
•
•
•
•
Examples:
Flipping a coin
Lottery number--pick 3
Lottery number--powerball
Rolling a die
Picking a team of players
Wheel of fortune
5-card draw poker
Independent events
• Spin the spinner 2 times.
(Note: 12 sectors)
• P(red, red)
• P(red, blue)
• P(blue, red)
B
Y
Y
RR W
R
R
B WW B
Dependent events
• This time, the spinner
can’t land on the same
space twice in a row.
But, it can land on the
same color twice in a row.
• P(red, red)
• P(red, blue)
• P(blue, red)
R
R
W
B
R
Y
R
Y
B WW B
Exploration 7.21
• Each group will be assigned a game. You
will make predictions, find the experimental
probability and the theoretical probabilities,
and then make a presentation to the class.
• You have 15 minutes to make your game
materials, and complete your part.
Expected value
• Expected value is used to determine
winnings. It is related to weighted averages
and probability.
• Think of this one: If I flip a coin and get a
head, I win $0.50. If I get a tail, I win nothing.
If I flip this coin twice, what do you think I
should expect to walk away with?
• If I flip 4 times, what will I expect to win?
• If I flip 100 times, … ?
• n times…?
Expected value
• In general, I consider each event that is
possible in my experiment. Each event
has it’s own consequence (win or lose
money, for example). And each event
has a probability associated with it.
• P(E1)•X1 + P(E2)•X2 + ••• + P(En)•Xn
Here are three easy
examples…
• Roll a 6-sided die. If you roll a “3”, then you
win $5.00. If you don’t roll a “3”, then you
have to pay $1.00.
• P(3) = 1/6
P(not 3) = 5/6
• P(3) • (5) + P(not 3) • (-1) =
• Expected Value
• (1/6)•(5) + (5/6)(-1) = 5/6 - 5/6 = 0.
• If the expected value is 0, we say the game
is fair.
Here are three easy
examples…
• Roll another die. If you roll a 3 or a 5, you
get a quarter. If you roll a 1, you get a dollar.
If you roll an even number, you pay 50¢.
P(3 or 5) = 1/3, P(1) = 1/6, P(even) = 1/2
Expected value
(1/3)•(.25) + (1/6)•(1) +(1/2)•(-.50) =
.0833 + .1667 -.25 = 0. Another fair game.
Here are three easy
examples…
• Is this grading system fair? There are
four choices on a multiple-choice
question. If you get the right answer,
you earn a point. If you get the wrong
answer, you lose a point.
• P(right answer)
P(wrong answer)
• Expected Value
Here’s a harder one…
• Suppose I spin the spinner.
RR W
B
• Here are the rules.
R
Y
If I spin blue or white, I get
R
Y
a quarter. If I spin red,
B
B
W
W
I get a nickel. If I spin
yellow, I have to pay 1 dollar.
• BLUE + WHITE + RED + YELLOW =
3/12 • .25 + 3/12 • .25 + 4/12 • .05 + 2/12 • (-1) =
.0625 + .0625 + .0167 + (-.1667) = -.025 or -2.5¢
Exploration 7.21
• Go back and redo the Native American
games. Compute the expected value
for each game in your groups. Turn
this in tomorrow.
Warm Up
• Review:
• Here is a data set.
5, 7, 18, 2, 0, 3, 8, 27, 9, 2, 28, 40, 6
• Find the mean, median, mode, and range.
• Make a stem and leaf plot for this data.
• Make a box and whisker plot for this data.
• 75% of this data is between ___ and ___ .
5, 7, 18, 2, 0, 3, 8, 27, 9,
2, 28, 40, 6
• Put in order:
• 0, 2, 2, 3, 5, 6, 7, 8, 9, 18, 27, 28, 40
• Mean: 11.9; Median: 7; Mode: 2
Range: 40
5, 7, 18, 2, 0, 3, 8, 27, 9,
2, 28, 40, 6
0
022356789
1
8
2
78
Key: 0/5 = 5
3
4
0
0
5 10
20
30
40
Agenda
• Review finding probability and expected
value
• Fundamental Counting Principle
• Combinations vs. Permutations
One event
• On a certain die, there are 3 fours, 2 fives,
and 1 six.
• P(rolling an odd) =
• P(rolling a number less than 6) =
• P(rolling a 6) =
• P(not rolling a 6) =
• P(rolling a 2) =
• Name two events that are complementary.
• Name two events that are mutually exclusive.
Two events
• I have 6 blue marbles and 4 red marbles in a
bag. If I do not replace the marbles, …
• P(blue) =
• P(red) =
• P(blue, blue) =
• P(red, blue) =
• P(blue, red) =
• Is this an example of independent or
dependent events?
Two events
• There are 8 girls and 7 boys in my class, who
want to be line leader or lunch helper, …
• P(G: LL, B: LH) =
• P(G: LL, G: LH) =
• P(B: LL, B: LH) =
• Is this an example of dependent or
independent events?
Watch the wording…
•
•
•
•
•
Suppose I flip a coin.
P(H) =
P(T) =
P(H or T) =
P(H and T) =
True/False
• Suppose you have a true/false section on
tomorrow’s exam. If there are 4 questions,…
• Make a list of all possibilities (tree diagram or
organized list).
• P(all 4 are true) =
• P(all 4 are false) =
• P(two are true and two are false) =
• Is this an example of independent or
dependent events?
Shortcut!
• If drawing a tree diagram takes too
long, consider this shortcut.
1st Q
2nd Q
3rd Q
4th Q
• Now, what do we do with these
numbers?
Fundamental Counting
Principle
• So, for the true/false scenario, it would be:
true or false for each question.
2 • 2 • 2 • 2 = 16 possible outcomes of the
true/false answers. Of course, only one of
these 16 is the correct outcome.
• So, if you guess, you will have a 1/16 chance
of getting a perfect score.
• Or, your odds for getting a perfect score are
1 : 15.
Fundamental Counting
Principle
• Suppose you have 5 multiple-choice
problems tomorrow, each with 4
choices. How many different ways can
you answer these problems?
• 4 • 4 • 4 • 4 • 4 = 1024
Fundamental Counting
Principle
• Now, suppose the question is
matching: there are 6 questions and 10
possible choices. Now, how many
ways can you match?
• 10 • 9 • 8 • 7 • 6 • 5 = 151,200
• How are true/false and multiple choice
questions different from matching
questions?
For dependent events, …
• Permutations vs. Combinations
• In a permutation, the order matters. In
a combination, the order does not
matter.
Examples
• I have 12 flowers, and I put 6 in a vase.
• I have 12 students, and I put 6 in a line.
• I have 12 identical math books, and I put 6
on a shelf.
• I have 12 different math books, and I put 6 on
a shelf.
• I have 12 more BINGO numbers to call, and I
call 6 more--then someone wins.
Permutations and
Combinations
• In a permutation, because order matters,
there are more outcomes to be considered
than in combinations.
• For example: if we have four students (A, B,
C, D), how many groups of 3 can we
choose?
• In a permutation, the group ABC is different
than the group CAB. In a combination, the
group ABC is the same as the group CAB.
Combinations: don’t count
duplicates
• So, how do I get rid of the duplicates?
• Let’s think.
• If I have two objects, A and B…
then my groups are AB and BA, or 2 groups.
• If I have three objects, A, B, and C…
then my groups are ABC, ACB, BAC, BCA,
CAB, CBA, or 6 groups.
• If I have three objects, A, B, and C…
then my groups are ABC, ACB, BAC, BCA,
CAB, CBA, or 6 groups.
• If I have 4 objects A, B, C, and D…
• Build from ABC:
DABC, ADBC, ABDC, ABCD
• Now build from ACB:
• DACB, ADCB, ACDB, ACBD
• How many possible? 6 • 4
Factorial
•
•
•
•
So, for 5 objects A, B, C, D, E, …
It will be 5 • 4 • 3 • 2 • 1.
We call this 5 factorial, and write it 5!
See how this is related to the Fundamental
Counting Principle?
So, if there are 5 objects to put in a row, then
there is 1 combination, but 120 permutations.
Two more practice
problems
• Suppose I have 16 kids on my team,
and I have to make up a starting line-up
of 11 kids.
• Permutation or combination: kids in the
field. Solve.
• Permutation or combination: kids
batting order. Solve.
• Kids in the field--the order of which kid
goes on the field first does not matter.
We just want a list of 9 kids from 16.
• 16 • 15 • 14 • 13 • 12 • 11 • 10 • 9 • 8
• Divide by 9! (to get rid of duplicates).
• Write it this way:
16 • 15 • 14 • 13 • 12 • 11 • 10 • 9 • 8
9 • 8 • 7 • 6 • 5 • 4 • 3 •2•1
• Combinations: 11440
• Permutations: 4,151,347,200
• Since the batting order does matter,
this is an example of a permutation.
Another example
• My bag of M&Ms has 4 blue, 3 green, 2
yellow, 4 red, and 8 browns--no orange.
• P(1st M&M is red)
• P(1st M&M is not brown)
• P(red, yellow)
• P(red, red)
• P(I eat the first 5 M&Ms in this order: blue,
blue, green, yellow, red)
• P(I gobble a handful of 2 blues, a green, a
yellow, and a red)
p. 488 #2, 8, 13
• 2. Candidates A, B, C, D:
4 • 3 • 2 • 1 = 24 or 4!
• 8. 9 players, 5 starting
9 • 8 • 7 • 6 • 5 = 15,120
• 13a. 2 • 4 • 3 = 24 choices
• 13b. If “no salad dressing” is a choice, then
5
•
4
•
6 = 120
appetizer main dish dessert.
Deal or no Deal
• You are a contestant on Deal or No
Deal. There are four amounts showing:
$5, $50, $1000, and $200,000. The
banker offers $50,000.
• Should you take the deal? Explain.
• How did the banker come up with
$50,000 as an offer?
A few practice problems
• A drawer contains 6 red socks and 3
blue socks.
P(pull 2, get a match)
P(pull 3, get 2 of a kind)
P(pull 4, all 4 same color)
• How many different license plates are
possible with 2 letters and 3 numbers?
(omit letters I, O, Q)
Is this an example of independent or
dependent events? Explain.
• If the mean of a sample is 12, and the
standard deviation is 3,
a. what interval contains about 68% of
the data?
b. what two intervals contain 50% of
the data?
c. How would the intervals change if
the standard deviation was 4?
True or false.
• A normal distribution is usually skewed to the
left.
• In a normal distribution, the mean and mode
are usually the same.
• A normal distribution describes numerical
data.
• A scatterplot shows categorical data.
• A histogram shows intervals of data,
• The median is easily determined with a box
and whisker plot.
Generalize the pattern.
• Find the perimeter of the nth term.
Generalize the pattern
• Find the number of squares in the nth
term.
Review Permutations and
Combinations
• I have 10 popsicles, and I give one to
Brendan each day for a week (7 days).
• How many ways can I do this?
• 10 • 9 • 8 • 7 • 6 • 5 • 4
• This is a permutation.
Review permutations and
combinations
• Janine’s boss has allowed her to have a
flexible schedule where she can work any
four days she chooses.
• How many schedules can Janine choose
from?
• 7•6•5•4
1•2•3•4
• Combination: working M,T,W,TH is the same
as working T,M,W,TH.
Last one
• Most days, you will teach Language Arts,
Math, Social Studies, and Science. If
Language Arts has to come first, how many
different schedules can you make?
• 1•3•2•1
• Permutation: the order of the schedule
matters.