Suppose I have two fair dice. Player one gets 2 points if the sum is odd. Player two gets 4 points...
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Transcript of Suppose I have two fair dice. Player one gets 2 points if the sum is odd. Player two gets 4 points...
![Page 1: 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?](https://reader036.fdocuments.net/reader036/viewer/2022062309/5697bfd21a28abf838cab8be/html5/thumbnails/1.jpg)
Suppose I have two fair Suppose I have two fair dice.dice.
Player one gets 2 points Player one gets 2 points if the sum is odd.if the sum is odd.
Player two gets 4 points Player two gets 4 points if the product is odd.if the product is odd.
Is this game fair?Is this game fair?
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Agenda• Review finding probability
• Determine expected value
• Is this game fair--1 player? 2 players?
• Fundamental Counting Principle
• Combinations vs. Permutations
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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…?
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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
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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.
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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.
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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
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Here’s a harder one…• Suppose I spin the spinner.• Here are the rules.
If I spin blue or white, I geta quarter. If I spin red,I get a nickel. If I spinyellow, 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¢
B
B B
R
R
R
R
Y
Y
W W
W
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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.
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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?
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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?
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Watch the wording…• Suppose I flip a coin.
• P(H) =
• P(T) =
• P(H or T) =
• P(H and T) =
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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?
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Shortcut!• If drawing a tree diagram takes too
long, consider this shortcut.
• Now, what do we do with these numbers?
1st Q 2nd Q 3rd Q 4th Q
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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.
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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
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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
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• How are true/false and multiple choice questions different from matching questions?
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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.
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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.
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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.
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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.
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• 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?
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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.
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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.
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• 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
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• Combinations: 11,440
• Permutations: 4,151,347,200
• Since the batting order does matter, this is an example of a permutation.
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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)
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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
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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?
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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)
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• 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.
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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.
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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.
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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.