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Review Homework pages 115-116
1. Example: Counting the number of heads in 10 coin tosses.
2.2/5
3.11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 54 55 56 61 62 63 64 65 66 both odd ¼
5. 61/100
10. 1 yellow, 2 red, 4 blue,5 green
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Bus
Plane Plane
Bus
Bus
BusPlane
Plane
Plane
PlaneCar
Car
Car
Car
Plane Bus
Plane CarBus Plane
Bus Bus
Bus Car
Car Plane
Car Bus
Car Car
Page 117 #5 - 7 #6 12
#7 ½ #5
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More Probability
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Sample Space and Events
Sample space( 표본 공간 ) : The collection of all possible outcomes for an experiment.
Event: A collection of outcomes for the experiment, that is, any subset of the sample space.
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Probability Notation
If E is an event, then P(E) stands for the probability that event E occurs. It is read “the probability of E”
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Venn diagram for event E
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Relationships Among Events
(not E): The event that “E does not occur.”
(A & B): The event that “both A and B occur.”
(A or B): The event that “either A or B or both occur.”
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The Classic Deck of 52 Playing Cards
• 4 Suits: Spades ♠, Hearts ♥, Clubs ♣, Diamonds ♦
• Each suit is made up of 13 cards or ranks.
• A (ace), 2, 3, 4, 5, 6, 7, 8, 9, 10, J (jack), Q (queen), K (king).
• Ace is usually considered high.
• J, Q, K are the face cards
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A Deck of Cards
Hearts Clubs Diamonds
Spades
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Sample space for rolling a die once
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Event (not E) where E is the probability of drawing a face card.
40/52=10/13
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Sample Space-1 Red Die, 1 Green Die - 36 Total Outcomes
Sample Space-Red Die, Green Die
1 2 3 4 5 6
1
2
3
4
5
6
Red DieGr
een D
ie
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The Sum of Two Die TossesSum Frequency 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2 12 1
What is the probability that the sum will be
5?
7?
What is the probability that the sum will be 10 or more?
What is the probability that the sum will be either 3 or less or 11 or more?
4/36=1/9
6/36=1/6
6/36=1/6
3/36 + 3/36=1/6
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Probabilities of 2 throws of the die
• What is the probability of a 1 and a 3?
• What is the probability of two sixes?
• What is the probability of at least one 3?
2/36
1/36
11/36
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Dice Roll by computer
http://www.shodor.org/interactivate/activities/ExpProbability/
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Two computer simulations of tossing a fair coin 100 times
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Law of Large Numbers
The greater the number of trials the more likely the experimental probability of an event will equal its theoretical probability.
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Tossing coins
http://www.shodor.org/interactivate/activities/Coin/
https://www.khanacademy.org/computer-programming/coin-flip-probability-simulator/1116198574
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Counting Principle
If there are m ways to do one thing, and n ways to do another, then there are m*n ways of doing both.
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Basic Properties of Probabilities
Property 1: The probability of an event is always between 0 and 1, inclusive.
Property 2: The probability of an event that cannot occur is 0. (An event that cannot occur is called an impossible event.)
Property 3: The probability of an event that must occur is 1. (An event that must occur is called a certain event.)
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A Deck of Cards
Hearts Clubs Diamonds
Spades
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The event the king of hearts is selected
1/52
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The event a king is selected
1/13 = 4/52
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The event a heart is selected
1/4 = 13/52
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What is the probability of an event that a face card is selected
3/13=12/52
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An event and its complement
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Homework
• Tree Diagram Worksheet