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Lecture 5: Chance Variability

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Title: Lecture 5: Chance Variability


1
Lecture 5 Chance Variability
  • The Law of Averages
  • The Expected Value and Standard Error
  • The Probability Histogram
  • The Normal Approximation for Probability
    Histograms
  • The Central Limit Theorem

2
How many heads?
  • Suppose you toss a fair coin 10,000 times.
    Roughly how many heads will you
    get?

3
Chance Fluctuations
  • Around 5,000, but not exactly. The actual
    number will fluctuate, due to chance. The chance
    error is
  • number of heads 5,000
  • How large is the chance error as the number of
    tosses increases indefinitely?

4
  • Kerrichs coing-tossing experiment.
    Chance error vs number of tosses.

5
  • Same data. Percentage chance error vs number
    of tosses.

6
The Law of Averages
  • As the number of tosses goes up,
  • the chance error is likely to get larger, but
    relative to the number of tosses, it gets
    smaller.
  • Equivalently, the percentage of heads is
    likely to become closer to 50, though not
    exactly.

7
Chance Processes
  • Coin tossing
  • Roulette
  • Sampling survey
  • Each experiment gives a different outcome,
    because of chance.

8
To what extent are the results influenced by
chance?
  • (a) find an analogy between the process and
    drawing numbers at random from a box.
  • (b) connect the variability of interest with
    the chance variability in the sum of the numbers
    drawn from the box.

9
The Sum of Draws
  • A shorthand for this process
  • (a) draw a number of tickets at random with
    replacement from a box of numbers.
  • (b) add the numbers on the tickets.
  • Simple, but will be used many times later.

10
Example 1
  • Roll a die 25 times, and take the sum. This is
    like the sum of 25 random draws with replacement
    from a box with tickets labeled 1 to 6.
  • Repeat this many times. You get a bunch of
    numbers (observed values) ranging from 25 to 150.

11
  • 10 computer simulations gave 88 84 80 90 83
    78 95 94 80 89. All between 75 and 100.
  • What is the chance that the sum is between 75 and
    100? This can be answered using the box model.

12
Making a Box Model
  • What numbers go into the box?
  • How many of each kind?
  • How many draws?

13
Las Vegas Roulette
  • A Las Vegas roulette wheel has 38 pockets 0, 00,
    1, 2,, 36. 18 are red, 18 are black. 0 and 00
    are green.
  • You bet 1 on red. If the ball lands on a red
    pocket, you get a 1 otherwise you lose 1.
    Consider your net gain in 10 games. What is the
    box model?

14
Solution
  • Box contains
  • 18 tickets labelled 1
  • 20 tickets labelled 1
  • 10 draws are made
  • (at random with replacement)

15
More Roulette
  • You bet number 7. You win 35 if the ball
    lands on 7, otherwise you lose 1. What are the
    tickets in the box?
  • 1 ticket labelled 35
  • 37 tickets labelled 1

16
A little Order in Chaos
  • A chance process is generating a long list of
    numbers. What can we say about it?
  • The numbers vary around the expected value, and
    the size of the chance error is measured by the
    standard error.

17
Expected Value of Sum
  • A box has 3 tickets labeled 1 and 1 ticket
    labeled 5.
  • Make n draws at random with replacement.
  • Every time you do this, you get a different sum.
    What is the EV?

18
  • About 3/4 of the n draws should give 1, and
    about 1/4 should give 5.
  • The EV for the sum of n draws is

19
EV of Sum Formula
  • But 2 is the average of the box, i.e., the
    average of the tickets.
  • For any box, the EV of the sum of n draws with
    replacement is
  • n x average of box

20
Standard Error of Sum
  • The sum of n draws hovers around the EV, and
    is likely to be off by a chance error about the
    size of the SE, given by
  • The explanation for this square-root law is
    beyond this course.

21
SE of sum and number of draws
  • It is intuitive that the larger the box SD,
    the more variable the sum. As the number of
    draws increases, the variability also goes up,
    but more slowly. For instance, the sum of 100
    draws is only 10 times as variable as a single
    draw.

22
Sum of 25 rolls of a die
  • Roll a die 25 times. The sum of 25 rolls varies
    if we repeat the experiment.
  • We expect the sum of 25 rolls to be around ___,
    give or take ___.
  • The EV and the SE are wanted here.

23
Solution
  • Box model tickets in box 1, 2, 3, 4,
    5, 6.
  • Average of box 3.5, SD of box 1.7.
  • EV 25 3.5 87.5
  • SE 5 1.7 8.5
  • So the sum of 25 rolls is around 87.5, give or
    take 8.5 or so.

24
Short-cut to compute SE
  • If there are only two kinds of tickets in the
    box, the SD can be found more easily. Let the
    two numbers be a and b, where a respective fractions be f and g. Then the SD is

25
Betting 7 on roulette
  • Box has 1 ticket labeled 35, 37 labeled ?1.
    Average of box ?0.05, SD of box 5.76.
  • You make 100 bets. EV ?5, SE 58. The big
    SE means you have some reasonable chance to win
    big, but on average you lose, and lose big.

26
SD and SE
  • SD measures the spread of a list of numbers.
  • SE measures the variability of a chance process.

27
Box Model, SD and SE
  • When drawing from a box, the SE of the sum is
    related the SD of the box by the square-root law.
  • Otherwise, the relationship is more
    complicated (e.g., the SE of the product).

28
Example
  • Box has tickets 0, 2, 3, 4, 6. What
    are the EV and SE of the sum of 25 draws at
    random with replacement?
  • Ave of box 3, SD of box 2. So
  • EV 25 3 75, SE 5 2 10.
  • The sum of 25 draws is likely to be around 75,
    give or take 10.

29
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30
Simulation vs Theory
  • Every number in the previous table is an
    observed value for the sum of the draws. The 100
    observed values differ from the EV 75 due to
    chance. Though the observed values range from 0
    to 150, the values in the table shows remarkably
    little spread. In fact, all except one of them
    are within 2.5 SEs of the EV.

31
Rule of Thumb
  • Observed values are rarely more than 2 or 3
    SEs away from the EV.

32
Chance and the Normal Curve
  • A large number of draws are made at random
    with replacement from a box. What is the chance
    that the sum will lie in a given range?
  • Convert to standard units using the EV and
    the SE and use the normal curve.

33
Box 0, 2, 3, 4, 6
  • 25 draws are made at random with replacement.
    This process is repeated many times.
  • About what percentage of the observed values
    should be between 50 and 100?
  • Same question Whats the chance that the sum
    will be between 50 and 100?

34
  • EV 75, SE 10.
  • 50 is ?2.5 in standard units, and 100 is 2.5 in
    SU.
  • The chance is roughly the area between ?2.5 and
    2.5 under the normal curve 99. In fact 99 of
    the 100 observed values in the table are between
    50 and 100.

35
  • About 68 should be between 65 and 85 in fact 73
    are.
  • About 95 should be between 55 and 95 98 are.
    The approximation is quite good.

36
Exercise
  • A die is rolled 60 times.
  • (a) The total number of spots should be
    around ____, give or take ____ or so.
  • (b) The number of 6s should be around ____,
    give or take ____ or so.

37
Solution (a)
  • Tickets are 1, 2, 3, 4, 5, 6.
    Total number of spots is like the sum of 60
    draws.
  • Average of box 3.5, SD of box 1.71.
  • EV 60 3.5 210, SE 60 1.71 13.

38
Solution (b)
  • EV 60 1/6 10.
  • For SE, need a box model.
  • Previous box is irrelevant. Instead of adding,
    each roll is classified is it a 6 or not? If
    6, add 1. Otherwise, add 0.
  • Tickets 0, 0, 0, 0, 0, 1. SD of box
    0.37. SE 60 0.37 3.

39
Adding or Classifying
  • Two operations on draws (1) adding, (2)
    classifying or counting.
  • EV and SE can be found by the same method, with
    the right box.
  • For (2), the box should have only 0s or 1s.
    1 is for what you want 0 for what you dont
    want.

40
Coin Tosses
  • Toss a coin 100 times. Estimate the chance of
    getting between 40 and 60 heads.
  • We are not summing, but counting the box has
    only 0s and 1s.
  • Since H and T are equally likely, it has one 0
    and one 1.

41
Coin Tosses
  • Average of box 0.5, SD of box 0.5.
  • EV 100 0.5 50, SE 10 0.5 5.
  • Chance is roughly the area under the normal curve
    between 2 and 2 95.

42
The Square-root Law and the Law of Averages
  • In n coin tosses, heads n/2 chance error.
  • Law of averages the chance error grows as n
    increases, but shrinks relative to n.
  • SE
  • The square-root law is the mathematical
    explanation of the law of averages.

43
What is the Chance?
  • Whats the chance that you get exactly 50 heads
    in 100 coin tosses? Exactly 51 heads? Between
    50 and 70 heads?
  • The binomial formula can be used, but is very
    tedious.
  • A good approximation is via the normal curve. To
    understand why this works, we need to look at
    probability histograms.

44
5 Coin Tosses
  • In 5 coin tosses, there are 25 32 equally
    likely outcomes.
  • number of heads number of
    patterns
  • 0
    1
  • 1
    5
  • 2
    10
  • 3
    10
  • 4
    5
  • 5
    1
  • A histogram for number of heads is a
    probability histogram.

45
Probability Histograms
  • A probability histogram of a chance process shows
    the chance of getting a particular outcome. The
    total area under a probability histogram is 100.
  • The EV and SE summarise the probability
    histogram, just as the Average and SD summarise
    the data histogram.

46
Constructing Probability Histograms
  • The probability histogram for the number of heads
    in 5 coin tosses is quite easy to make.
  • For more complicated chance processes, a computer
    can be used.

47
A Pair of Dice
  • The computer is programmed to roll a pair of dice
    and record the total number of spots. This is
    repeated many times.
  • Make a histogram of the simulated data. As the
    number of repetition becomes large, the data
    histogram converges to the probability histogram.

48
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49
Another Probability Histogram
  • A similar method can be used to get the
    probability histogram for the product of the
    number of spots on a pair of dice.

50
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51
Probability Histograms and the Normal Curve
  • The probability histogram for the number of heads
    in 100 coin tosses is symmetric, and sort of fits
    the normal curve. The fit gets better with 400,
    and 900 tosses.
  • The conclusion if the number of tosses is large
    enough, the probability histogram is close to the
    normal curve.

52
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53
The Normal Approximation
  • A coin will be tossed 100 times. Estimate the
    chance of getting
  • (a) exactly 50 heads.
  • (b) between 45 and 55 heads inclusive.
  • (c) between 45 and 55 heads exclusive.

54
Solution
  • EV 50, SE 5.
  • (a) The chance is the area of the block above 50
    in the probability histogram. This block goes
    from 49.5 to 50.5. In SU, they become ?0.1 and
    0.1. The chance is approximately the area under
    the normal curve between ?0.1 and 0.1 7.97.
    Actual 7.96.

55
  • (b) Chance area under the probability histogram
    between 44.5 and 55. This is approximately the
    area under the normal curve between ?1.1 and 1.1
    72.87. Actual 72.87.
  • (c) The area between 45.5 and 54.5 is about that
    under the normal curve between ?0.9 and 0.9
    63.19. Actual 63.18.

56
Continuity Correction
  • The continuity correction was used. For
    example, in finding the chance that the number of
    heads is between 45 and 55 inclusive, we look at
    the area under the probability histogram between
    44.5 and 55.5.

57
Continuity Correction
  • If you are not told whether the endpoints are
    included, then a compromise is not to do a
    correction.
  • For example, the chance that the number of heads
    is between 45 and 55 is approximated by the area
    under the probability histogram between 45 and
    55.

58
Drawing from a Box
  • The number of heads in a coin-tossing process is
    like the sum of repeated draws from a box with
    tickets 0 and 1.
  • How about other boxes? A lopsided box with 9
    0s and 1 1.

59
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60
Lopsided Box
  • With 25 draws, the probability histogram is a lot
    higher than the normal curve on the left, and
    lower on the right the normal approximation is
    not so good.
  • With 100 draws, the approximation is much better.
    Even better with 400 draws.

61
Another Box
  • A box with three tickets 1, 2 and 3. This
    box is symmetric, so even with 25 draws, the
    normal approximation is quite good.

62
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63
Last Box
  • A box with three tickets 1, 2 and 9. The
    histogram of the tickets does not look normal at
    all.
  • But with enough draws, say 100, the approximation
    is excellent.

64
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65
How about the product?
  • Roll a die 10 times, and record the product of
    the 10 numbers.
  • Does the probability histogram for the product
    converge to the normal curve, as we increase the
    number of rolls?

66
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67
The Central Limit Theorem
  • When the number of draws from a box is large, the
    probability histogram for the sum is well
    approximated by the normal curve.
  • This mathematical fact holds for ANY box, and
    plays an important role in many statistical
    procedures. Hence the name.

68
How many draws?
  • How many draws is enough for the approximation
    to be good? It depends on the box. However, for
    many boxes, the probability histogram for the sum
    of 100 draws will be close enough to the normal
    curve.

69
Probability Histogram, EV, SE
  • When the probability histogram is close to the
    normal curve, it is completely determined by the
    EV and SE. The EV gives the centre, and the SE
    tells you the spread of the probability histogram.

70
Two Convergences
  • The first convergence has to do with constructing
    a single probability histogram. Here, the number
    of draws is fixed, and the process is repeated
    many times. The data histogram converges to the
    probability histogram.
  • This applies not just to sums, e.g., products.

71
Two Convergences (cont.)
  • The second kind concerns the behaviour of
    probability histograms as the number of draws
    goes to infinity. Here the Central Limit Theorem
    says that the probability histogram converges to
    the normal curve.
  • The CLT only applies to sums.

72
Two Convergences (cont.)
  • In the first convergence, the chance process is
    fixed (e.g., number of draws), and the number
    repetitions goes to infinity.
  • In the second convergence, the number of draws
    contributing to the sum goes to infinity.

73
Normal Approximation for Data
  • When the number of repetitions (samples) is
    large, the first convergence ensures that the
    data histogram is close to the probability
    histogram.
  • When the number of draws is large, the
    probability histogram is close to the normal
    curve.
  • When both are large, the data histogram is well
    approximated by the normal curve. This can be
    proved mathematically.

74
Applying the Theory
  • Is the data-generating process like taking the
    sum of numbers drawn from a box?
  • If so, then when the number of draws are large,
    you can make very precise predictions.
  • Establishing this analogy is beyond mathematics.
    It is a question of fact.
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