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Probability

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Title: Probability


1
Chapter 3
  • Probability

2
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?

3
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.

4
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 ?

5
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.

6
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?

7
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

8
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!

9
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 properties of
    probabilities, such as independence.

10
Probability
  • I toss a fair coin (where fair means equally
    likely outcomes)
  • What are the possible outcomes?
  • Head and tail This is called a dichotomous
    experiment because it has only two possible
    outcomes. S H,T.
  • 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?
  • ?????

11
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

12
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)

13
Where do probabilities come from?
  • Probabilities from data
  • or Empirical probabilities
  • What is the probability that a randomly selected
    WSU student regularly drinks alcohol?
  • 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 or 60.9 chance)

14
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.

15
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 S AH, 2H, 3H, , KH,, AS,,KS

16
Simple Probability Models
  • an event is a collection of outcomes
  • E.g. E 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 E is denoted as E
    , it contains all outcomes not in E E.g. E
    card drawn is not a heart
  • card drawn is a spade, club or
    diamond

17
Simple Probability Models
  • The probability that an event E occurs
  • is written in shorthand as P(E).

18
House Sales Example
  • Below is a table containing some information for
    a sample of 600 sales of single family houses in
    1999.

19
House Sales Example
  • Let A be the event that a sale is over
    400,000
  • A is the event that a sale is NOT over 400,000

20
House Sales Example
  • B be the event that a sale is made
    within 45 days
  • So B is the event that a sale takes longer than
    45 days

21
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
22
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. A occurs.

P(A) (155384)/600 539/600 0.898 Note
that P(A) P(A) 1 and that P(A) 1 P(A)
23
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. B occurs.

P(B) (300 54)/600 354/600 0.59
24
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
25
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.

26
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
27
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

28
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
29
Conditional Probability
  • The sample space is reduced.
  • Key words that indicate conditional probability
    are
  • given that, of those, if , assuming
    that

30
Conditional Probability
  • The probability of event E occurring given that
    event F has already occurred
  • is written in shorthand as P(EF)

31
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?

32
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
33
Independence
  • Events E and F are said to independent if

P(EF) P(E)
For the house sales data the P(A) 61/600
.102 and we have just seen P(AB) .081 thus it
seems that A and B are NOT independent.
34
1. Heart Disease
  • In 1996, 6631 Minnesotans died from coronary
    heart disease. The numbers of deaths classified
    by age and gender are

35
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

36
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
37
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.

38
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
39
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
40
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.

41
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(CB) (4992176)/2904 0.9211
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