Patrick's%20Casino

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Patrick's Casino
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What is the probability of picking an ace?
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Probability
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What is the probability of picking an ace? 4 / 52
.077 or 7.7 chances in 100
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Every card has the same probability of being
picked
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What is the probability of getting a 10, J, Q, or
K?
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(.077) (.077) (.077) (.077) .308 16 / 52
.308
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What is the probability of getting a 2 and then
after replacing the card getting a 3 ?
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(.077) (.077) .0059
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What is the probability that the two cards you
draw will be a black jack?
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10 Card (.077) (.077) (.077) (.077)
.308 Ace after one card is removed 4/51
.078 (.308)(.078) .024
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Practice

• What is the probability of rolling a 1 using a
  six sided dice?
• What is the probability of rolling either a 1
  or a 2 with a six sided dice?
• What is the probability of rolling two 1s
  using two six sided dice?

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Practice

• What is the probability of rolling a 1 using a
  six sided dice?
• 1 / 6 .166
• What is the probability of rolling either a 1
  or a 2 with a six sided dice?
• What is the probability of rolling two 1s
  using two six sided dice?

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Practice

• What is the probability of rolling a 1 using a
  six sided dice?
• 1 / 6 .166
• What is the probability of rolling either a 1
  or a 2 with a six sided dice?
• (.166) (.166) .332
• What is the probability of rolling two 1s
  using two six sided dice?

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Practice

• What is the probability of rolling a 1 using a
  six sided dice?
• 1 / 6 .166
• What is the probability of rolling either a 1
  or a 2 with a six sided dice?
• (.166) (.166) .332
• What is the probability of rolling two 1s
  using two six sided dice?
• (.166)(.166) .028

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Cards

• What is the probability of drawing an ace?
• What is the probability of drawing another ace?
• What is the probability the next four cards you
  draw will each be an ace?
• What is the probability that an ace will be in
  the first four cards dealt?

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Cards

• What is the probability of drawing an ace?
• 4/52 .0769
• What is the probability of drawing another ace?
• 4/52 .0769 3/51 .0588 .0769.0588 .0045
• What is the probability the next four cards you
  draw will each be an ace?
• .0769.0588.04.02 .000003
• What is the probability that an ace will be in
  the first four cards dealt?
• .0769.078.08.082 .3169

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Probability
1.00
.00
Event must occur
Event will not occur
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Probability

• In this chapter we deal with discreet variables
• i.e., a variable that has a limited number of
  values
• Previously we discussed the probability of
  continuous variables (Z scores)
• It does not make sense to seek the probability of
  a single score for a continuous variable
• Seek the probability of a range of scores

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Key Terms

• Independent event
• When the occurrence of one event has no effect on
  the occurrence of another event
• e.g., voting behavior, IQ, etc.
• Mutually exclusive
• When the occurrence of one even precludes the
  occurrence of another event
• e.g., your year in the program, if you are in
  prosem

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Key Terms

• Joint probability
• The probability of the co-occurrence of two or
  more events
• The probability of rolling a one and a six
• p (1, 6)
• p (Blond, Blue)

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Key Terms

• Conditional probabilities
• The probability that one event will occur given
  that some other vent has occurred
• e.g., what is the probability a person will get
  into a PhD program given that they attended
  Villanova
• p(Phd l Villa)
• e.g., what is the probability that a person will
  be a millionaire given that they attended college
• p( l college)

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Example
Owns a video game Does not own a video game Total
No Children 10 35 45
Children 25 30 55
Total 35 65 100
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What is the simple probability that a person will
own a video game?
Owns a video game Does not own a video game Total
No Children 10 35 45
Children 25 30 55
Total 35 65 100
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What is the simple probability that a person will
own a video game? 35 / 100 .35
Owns a video game Does not own a video game Total
No Children 10 35 45
Children 25 30 55
Total 35 65 100
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What is the conditional probability of a person
owning a video game given that he or she has
children? p (video l child)
Owns a video game Does not own a video game Total
No Children 10 35 45
Children 25 30 55
Total 35 65 100
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What is the conditional probability of a person
owning a video game given that he or she has
children?25 / 55 .45
Owns a video game Does not own a video game Total
No Children 10 35 45
Children 25 30 55
Total 35 65 100
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What is the joint probability that a person will
own a video game and have children? p(video,
child)
Owns a video game Does not own a video game Total
No Children 10 35 45
Children 25 30 55
Total 35 65 100
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What is the joint probability that a person will
own a video game and have children? 25 / 100 .25
Owns a video game Does not own a video game Total
No Children 10 35 45
Children 25 30 55
Total 35 65 100
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25 / 100 .25.35 .55 .19
Owns a video game Does not own a video game Total
No Children 10 35 45
Children 25 30 55
Total 35 65 100
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The multiplication rule assumes that the two
events are independent of each other it does
not work when there is a relationship!
Owns a video game Does not own a video game Total
No Children 10 35 45
Children 25 30 55
Total 35 65 100
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Practice
Republican Democrat Total
Male 52 27 79
Female 18 65 83
Total 70 92 162
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p (republican) p(female)p (republican,
male) p(female, republican)p (republican l
male) p(male l republican)
Republican Democrat Total
Male 52 27 79
Female 18 65 83
Total 70 92 162
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p (republican) 70 / 162 .43p (republican,
male) 52 / 162 .32p (republican l male) 52
/ 79 .66
Republican Democrat Total
Male 52 27 79
Female 18 65 83
Total 70 92 162
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p(female) 83 / 162 .51p(female, republican)
18 / 162 .11p(male l republican) 52 / 70
.74
Republican Democrat Total
Male 52 27 79
Female 18 65 83
Total 70 92 162
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Foot Race

• Three different people enter a foot race
• A, B, C
• How many different combinations are there for
  these people to finish?

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Foot Race

• A, B, C
• A, C, B
• B, A, C
• B, C, A
• C, B, A
• C, A, B
• 6 different permutations of these three names
  taken three at a time

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Foot Race

• Six different people enter a foot race
• A, B, C, D, E, F
• How many different permutations are there for
  these people to finish?

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Permutation

• Ingredients
• N total number of events
• r number of events selected

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Permutation

• Ingredients
• N total number of events
• r number of events selected
• A, B, C, D, E, F Note 0! 1

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Foot Race

• Six different people enter a foot race
• A, B, C, D, E, F
• How many different permutations are there for
  these people to finish in the top three?
• A, B, C A, C, D A, D, E B, C, A

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Permutation

• Ingredients
• N total number of events
• r number of events selected

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Permutation

• Ingredients
• N total number of events
• r number of events selected

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Foot Race

• Six different people enter a foot race
• If a person only needs to finish in the top three
  to qualify for the next race (i.e., we dont care
  about the order) how many different outcomes are
  there?

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Combinations

• Ingredients
• N total number of events
• r number of events selected

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Combinations

• Ingredients
• N total number of events
• r number of events selected

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Note

• Use Permutation when ORDER matters
• Use Combination when ORDER does not matter

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Practice

• There are three different prizes
• 1st 1,00
• 2nd 500
• 3rd 100
• There are eight finalist in a drawing who are
  going to be awarded these prizes.
• A person can only win one prize
• How many different ways are there to award these
  prizes?

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Practice

• 336 ways of awarding the three different prizes

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Practice

• There are three prizes (each is worth 200)
• There are eight finalist in a drawing who are
  going to be awarded these prizes.
• A person can only win one prize
• How many different ways are there to award these
  prizes?

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Combinations

• 56 different ways to award these prizes

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Practice

• A shirt comes in four sizes and six colors. One
  also has the choice of a dragon, alligator, or no
  emblem on the pocket. How many different kinds
  of shirts can you order?

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Practice

• A shirt comes in four sizes and six colors. One
  also has the choice of a dragon, alligator, or no
  emblem on the pocket. How many different kinds
  of shirts can you order?
• 463 72
• Dont make it hard on yourself!

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Practice

• In a California Governor race there were 135
  candidates. The state created ballots that would
  list candidates in different orders. How many
  different types of ballots did the state need to
  create?

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Practice

• 2.6904727073180495e230
• Or

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• 26,904,727,073,180,495,000,000,000,000,000,000,00
  0,000,000,000,000,000,000,000,000,000,000,000,000,
  000,000,000,000,000,000,000,000,000,000,000,000,00
  0,000,000,000,000,000,000,000,000,000,000,000,000,
  000,000,000,000,000,000,000,000,000,000,000,000,00
  0,000,000,000,000,000,000,000,000,000,000,000,000,
  000,000,000

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Bonus Points

• Suppose youre on a game show and youre given
  the choice of three doors. Behind one door is a
  car behind the others, goats. The car and the
  goats were placed randomly behind the doors
  before the show. The rules of the game show are
  as follows After you have chosen a door, the
  door remains closed for the time being. The game
  show host, Monty Hall, who knows what is behind
  the doors, now has to open one of the two
  remaining doors, and the door he opens must have
  a goat behind it. If both remaining doors have
  goats behind them, he chooses one randomly. After
  Monty Hall opens a door with a goat, he will ask
  you to decide whether you want to stay with your
  first choice or to switch to the last remaining
  door. Imagine that you chose Door 1 and the host
  opens Door 3, which has a goat. He then asks you
  Do you want to switch to Door Number 2? Is it
  to your advantage to change your choice? 

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• http//www.nytimes.com/2008/04/08/science/08monty.
  html?_r1

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You pick 1
Door 1 Door 2 Door 3 Results
GAME 1 AUTO GOAT GOAT Switch and you lose.
GAME 2 GOAT AUTO GOAT Switch and you win.
GAME 3 GOAT GOAT AUTO Switch and you win.
GAME 4 AUTO GOAT GOAT Stay and you win.
GAME 5 GOAT AUTO GOAT Stay and you lose.
GAME 6 GOAT GOAT AUTO Stay and you lose.
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Practice

• The probability of winning Blingoo is .30
• What is the probability that you will win 20 of
  the next 30 games of Blingoo ?
• Note previous probability methods do not work
  for this question

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Binomial Distribution

• Used with situations in which each of a number of
  independent trials results in one of two mutually
  exclusive outcomes

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Binomial Distribution

• Ingredients
• N total number of events
• p the probability of a success on any one
  trial
• q (1 p) the probability of a failure on
  any one trial
• X number of successful events

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Game of Chance

• The probability of winning Blingoo is .30
• What is the probability that you will win 20 of
  the next 30 games of Blingoo ?
• Note previous probability methods do not work
  for this question

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Binomial Distribution

• Ingredients
• N total number of events
• p the probability of a success on any one
  trial
• q (1 p) the probability of a failure on
  any one trial
• X number of successful events

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Binomial Distribution

• Ingredients
• N total number of events
• p the probability of a success on any one
  trial
• q (1 p) the probability of a failure on
  any one trial
• X number of successful events

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Binomial Distribution

• Ingredients
• N total number of events
• p the probability of a success on any one
  trial
• q (1 p) the probability of a failure on
  any one trial
• X number of successful events

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Binomial Distribution

• Ingredients
• N total number of events
• p the probability of a success on any one
  trial
• q (1 p) the probability of a failure on
  any one trial
• X number of successful events

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Binomial Distribution

• Ingredients
• N total number of events
• p the probability of a success on any one
  trial
• q (1 p) the probability of a failure on
  any one trial
• X number of successful events

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Binomial Distribution

• Ingredients
• N total number of events
• p the probability of a success on any one
  trial
• q (1 p) the probability of a failure on
  any one trial
• X number of successful events

p .000029
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What does this mean?

• p .000029
• This is the probability that you would win
  EXACTLY 20 out of 30 games of Blingoo

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Game of Chance

• Playing perfect black jack the probability of
  winning a hand is .498
• What is the probability that you will win 8 of
  the next 10 games of blackjack?

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Binomial Distribution

• Ingredients
• N total number of events
• p the probability of a success on any one
  trial
• q (1 p) the probability of a failure on
  any one trial
• X number of successful events

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Binomial Distribution

• Ingredients
• N total number of events
• p the probability of a success on any one
  trial
• q (1 p) the probability of a failure on
  any one trial
• X number of successful events

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Binomial Distribution

• Ingredients
• N total number of events
• p the probability of a success on any one
  trial
• q (1 p) the probability of a failure on
  any one trial
• X number of successful events

p .0429
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Excel
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Binomial Distribution

• What is this doing?
• Its combining together what you have learned so
  far!
• One way to fit our 8 wins would be (joint
  probability)
• W, W, W, W, W, W, W, W, L, L
• (.498)(.498)(.498)(.498)(.498)(.498)(.498)(.498)(.
  502)(.502)
• (.4988)(.5022).00095
• pX q(N-X)

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Binomial Distribution

• Ingredients
• N total number of events
• p the probability of a success on any one
  trial
• q (1 p) the probability of a failure on
  any one trial
• X number of successful events

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Binomial Distribution

• Ingredients
• N total number of events
• p the probability of a success on any one
  trial
• q (1 p) the probability of a failure on
  any one trial
• X number of successful events

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Binomial Distribution

• Other ways to fit our question
• W, L, L, W, W, W, W, W
• L, W, W, W, W, L, W, W
• W, W, W, L, W, W, W, L
• L, L, W, W, W, W, W, W
• W, L, W, L, W, W, W, W
• W, W, L, W, W, W, L, W

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Binomial Distribution

• Other ways to fit our question
• W, L, L, W, W, W, W, W .00095
• L, W, W, W, W, L, W, W .00095
• W, W, W, L, W, W, W, L .00095
• L, L, W, W, W, W, W, W .00095
• W, L, W, L, W, W, W, W .00095
• W, W, L, W, W, W, L, W .00095
• Each combination has the same probability but
  how many combinations are there?

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Combinations

• Ingredients
• N total number of events
• r number of events selected
• 45 different combinations

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Binomial Distribution

• Ingredients
• N total number of events
• p the probability of a success on any one
  trial
• q (1 p) the probability of a failure on
  any one trial
• X number of successful events

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Binomial Distribution

• Any combination would work
• . 00095 00095 00095 00095 00095 00095
  00095 00095. . . . . . 00095
• Or 45 . 00095 .04

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Binomial Distribution

• Ingredients
• N total number of events
• p the probability of a success on any one
  trial
• q (1 p) the probability of a failure on
  any one trial
• X number of successful events

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Practice

• You bought a ticket for a fire department lottery
  and your brother has bought two tickets. You just
  read that 1000 tickets were sold.
• a) What is the probability you will win the grand
  prize?
• b) What is the probability that your brother will
  win?
• c) What is the probably that you or your bother
  will win?

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5.2

• A) 1/1000 .001
• B)2/1000 .002
• C) .001 .002 .003

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Practice

• Assume the same situation at before except only
  a total of 10 tickets were sold and there are two
  prizes.
• a) Given that you didnt win the first prize,
  what is the probability you will win the second
  prize?
• b) What is the probability that your borther will
  win the first prize and you will win the second
  prize?
• c) What is the probability that you will win the
  first prize and your brother will win the second
  prize?
• d) What is the probability that the two of you
  will win the first and second prizes?

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5.3

• A) 1/9 .111
• B) 2/10 1/9 (.20)(.111) .022
• C) 1/10 2/9 (.10)(.22) .022
• D) .022 .022 .044

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Practice

• In some homes a mothers behavior seems to be
  independent of her baby's, and vice versa. If
  the mother looks at her child a total of 2 hours
  each day, and the baby looks at the mother a
  total of 3 hours each day, and if they really do
  behave independently, what is the probability
  that they will look at each other at the same
  time?

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5.8

• 2/24 .083
• 3/24 .125
• .083.125 .01

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Practice

• Abe ice-cream shot has six different flavors of
  ice cream, and you can order any combination of
  any number of them (but only one scoop of each
  flavor). How many different ice-cream cone
  combinations could they truthfully advertise
  (note, we dont care about the order of the
  scoops and an empty cone doesnt count).

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5.29

6 15 20 15 6 1 63
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Extra Brownie Points!

• Lottery
• To Win
• choose the 5 winnings numbers
• from 1 to 49
• AND
• Choose the "Powerball" number
• from 1 to 42
• What is the probability you will win?
• Use combinations to answer this question