Download Extra 7.1, 7.2

Suppose I have two fair
dice.
Player one gets 2 points
if the sum is odd.
Player two gets 4 points
if the product is odd.
Is this game fair?
Agenda
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Review finding probability
Determine expected value
Is this game fair--1 player? 2 players?
Fundamental Counting Principle
Combinations vs. Permutations
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¢
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 disjoint.
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…
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•
•
•
•
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.
• I have 18 cans of soda: 3 diet pepsi, 4 diet
coke, 5 pepsi, and 6 sprite.
• Permutation or combination?
– I pick 4 cans of soda randomly.
– I give 4 friends each one can of soda, randomly.
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, BAC, ACB, 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
•
Keep going… How many possible?
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 9 kids.
• Permutation or combination: kids in the field
(don’t consider the position). 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: 11,440
• 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)
Homework
• Due on Tuesday: do all, turn in the
bold.
• Section 7.4 p. 488
#2, 3, 7, 8, 12, 13, 15
• Read section 8.1
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.
Review Permutations and
Combinations
• I have 10 different flavored 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.