Probabilities and Proportions

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Probabilities and Proportions
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Lotto

• I am offered two lotto cards
• Card 1 has numbers
• Card 2 has numbers
• Which card should I take so that I have the
  greatest chance of winning lotto?

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Roulette

• In the casino I wait at the roulette wheel until
  I see a run of at least five reds in a row.
• I then bet heavily on a black.
• I am now more likely to win.

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Coin Tossing

• I am about to toss a coin 20 times.
• What do you expect to happen?
• Suppose that the first four tosses have been
  heads and there are no tails so far. What do you
  expect will have happened by the end of the 20
  tosses ?

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Coin Tossing

• Option A
• Still expect to get 10 heads and 10 tails. Since
  there are already 4 heads, now expect to get 6
  heads from the remaining 16 tosses. In the next
  few tosses, expect to get more tails than heads.
• Option B
• There are 16 tosses to go. For these 16 tosses I
  expect 8 heads and 8 tails. Now expect to get 12
  heads and 8 tails for the 20 throws.

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TV Game Show

• In a TV game show, a car will be given away.
• 3 keys are put on the table, with only one of
  them being the right key. The 3 finalists are
  given a chance to choose one key and the one who
  chooses the right key will take the car.
• If you were one of the finalists, would you
  prefer to be the 1st, 2nd or last to choose a key?

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Lets Make a Deal Game Show

• You pick one of three doors
• two have booby prizes behind them
• one has lots of money behind it
• The game show host then shows you a booby prize
  behind one of the other doors
• Then he asks you Do you want to change doors?
• Should you??! (Does it matter??!)
• See the following website
• http//www.stat.sc.edu/west/javahtml/LetsMakeaDea
  l.html

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Game Show Dilemma

• Suppose you choose door A. In which case Monty
  Hall will show you either door B or C depending
  upon what is behind each.
• No Switch Strategy here is what happens

Result A B C
Win Car Goat Goat
Lose Goat Car Goat
Lose Goat Goat Car
P(WIN) 1/3
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Game Show Dilemma

• Suppose you choose door A, but ultimately
  switch. Again Monty Hall will show you either
  door B or C depending upon what is behind each.
• Switch Strategy here is what happens

Monty will show either B or C. You switch to the
one not shown and lose.
Monty will show door C, you switch to B and win.
Result A B C
Lose Car Goat Goat
Win Goat Car Goat
Win Goat Goat Car
Monty will show door B, you switch to C and win.
P(WIN) 2/3 !!!!
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Matching Birthdays

• In a room with 23 people what is the probability
  that at least two of them will have the same
  birthday?
• Answer .5073 or 50.73 chance!!!!!
• How about 30?
• .7063 or 71 chance!
• How about 40?
• .8912 or 89 chance!
• How about 50?
• .9704 or 97 chance!

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Probability

• What is Chapter 3 trying to do?
• Introduce us to basic ideas about probabilities
• what they are and where they come from
• simple probability models
• conditional probabilities
• independent events
• Teach us how to calculate probabilities
• through tables of counts and probability tables
  for independent events

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Probability

• I toss a fair coin (where fair means equally
  likely outcomes)
• What are the possible outcomes?
• Head and tail
• What is the probability it will turn up heads?
• 1/2
• I choose a person at random and check which eye
  she/he winks with
• What are the possible outcomes?
• Left and right
• What is the probability they wink with their left
  eye?
• ?????

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What are Probabilities?

• A probability is a number between 0 1 that
  quantifies uncertainty
• A probability of 0 identifies impossibility
• A probability of 1 identifies certainty

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Where do probabilities come from?

• Probabilities from models
• The probability of getting a four when a fair
  dice is rolled is
• 1/6 (0.1667 or 16.7)

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Where do probabilities come from?

• Probabilities from data
• In a survey conducted by students in a STAT 110
  course there were 348 WSU students sampled.
• 212 of these students stated they regularly drink
  alcohol.
• The estimated probability that a randomly chosen
  Winona State students drinks alcohol is
• 212/348 (0.609, 60.9)

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Where do probabilities come from?

• Subjective Probabilities
• The probability that there will be another
  outbreak of ebola in Africa within the next year
  is 0.1.
• The probability of rain in the next 24 hours is
  very high. Perhaps the weather forecaster might
  say a there is a 70 chance of rain.
• A doctor may state your chance of successful
  treatment.

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Simple Probability Models

• Terminology
• a random experiment is an experiment whose
  outcome cannot be predicted
• E.g. Draw a card from a well-shuffled pack
• a sample space is the collection of all possible
  outcomes
• 52 outcomes (AH, 2H, 3H, , KH,, AS,
  ,KS)

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Simple Probability Models

• an event is a collection of outcomes
• E.g. A card drawn is a heart
• an event occurs if any outcome making up that
  event occurs
• drawing a 5 of hearts
• the complement of an event A is denoted as ,
  it contains all outcomes not in A Eg card
  drawn is not a heart
• card drawn is a spade, club or
  diamond

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Simple Probability Models

• The probability that an event A occurs
• is written in shorthand as P(A).

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House Sales Example

• Below is a table containing some information for
  a sample of 600 sales of single family houses in
  1999.

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House Sales Example

• Let A be the event that a sale is over
  400,000
• is the event that a sale is NOT over 400,000

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House Sales Example

• B be the event that a sale is made
  within 45 days
• So is the event that a sale takes longer than
  45 days

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House Sales Example

• For a sale selected at random from these 600
  sales, find the probability that the sale was
• over 400,000, i.e. event A occurs.

P(A) 61/600 0.102
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House Sales Example

• For a sale selected at random from these 600
  sales, find the probability that the sale was
• not over 400,000, i.e. occurs.

P( ) (155384)/600 539/600 0.898 Note
that P(A) P( ) 1 and that P( ) 1 P(A)
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House Sales Example

• For a sale selected at random from these 600
  sales, find the probability that the sale was
• c) made in 45 days or more, i.e. occurs.

P( ) (300 54)/600 354/600 0.59
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House Sales Example

• For a sale selected at random from these 600
  sales, find the probability that the sale was
• d) made within 45 days and sold for over
  400,000, i.e. both B and A occur.

P(B and A) 20/600 0.033
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House Sales Example

• For a sale selected at random from these 600
  sales, find the probability that the sale was
• e) made within 45 days and/or sold for over
  400,000, i.e. either A or B occur.

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House Sales Example

• For a sale selected at random from these 600
  sales, find the probability that the sale was
• e) made within 45 days and/or sold for over
  400,000.

P(B and/or A) (246 61 20)/600 287/600
0.478
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House Sales Example

• For a sale selected at random from these 600
  sales, find the probability that the sale was
• f ) on the market for less than 45 days given
  that it sold for over 400,000

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House Sales Example

• For a sale selected at random from these 600
  sales, find the probability that the sale was
• f ) on the market for less than 45 days given
  that it sold for over 400,000

P(B given A) P(BA) 20/61 0.328
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Conditional Probability

• The sample space is reduced.
• Key words that indicate conditional probability
  are
• given that, of those, if , assuming
  that

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Conditional Probability

• The probability of event A occurring given that
  event B has already occurred
• is written in shorthand as P(AB)

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House Sales Example

• For a sale selected at random from these 600
  sales,
• g) What proportion of the houses that sold in
  less than 45 days, sold for more than 400,000?

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House Sales Example

• For a sale selected at random from these 600
  sales,
• g) What proportion of the houses that sold in
  less than 45 days, sold for more than 400,000?

P (AB) 20/246 0.081
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1. Heart Disease

• In 1996, 6631 New Zealanders died from coronary
  heart disease. The numbers of deaths classified
  by age and gender are

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1. Heart Disease

• Let
• A be the event of being under 45
• B be the event of being male
• C be the event of being over 64

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1. Heart Disease

• Find the probability that a randomly chosen
  member of this population at the time of death
  was
• under 45

P(A) 92/6631 0.0139
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1. Heart Disease

• Find the probability that a randomly chosen
  member of this population at the time of death
  was
• male assuming that the person was younger than
  45.

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Heart Disease

• Find the probability that a randomly chosen
  member of this population at the time of death
  was
• male given that the person was younger than 45.

P(BA) 79/92 0.8587
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1. Heart Disease

• Find the probability that a randomly chosen
  member of this population at the time of death
  was
• c) male and was over 64.

P(B and C) (1081 1795)/6631 2876/6631
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1. Heart Disease

• Find the probability that a randomly chosen
  member of this population at the time of death
  was
• d) over 64 given they were female.

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1. Heart Disease

• Find the probability that a randomly chosen
  member of this population at the time of death
  was
• d) over 64 given they were female.

P(C B) (4992176)/2904 0.9211