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Ch. 11 The Law of Averages

Ch. 11 The Law of Averages

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Ch. 11 The Law of Averages

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  1. Ch. 11 The Law of Averages The Law of Averages (Law of Large Numbers) The Stochastic Processes Making a Box Model

  2. 2 3 The Stochastic Processes Making a Box Model INDEX The Law of Averages (Law of Large Numbers) 1

  3. \100 is tossed 1. The Law of Averages The Law of Averages (1) Head Tail Heads with probability 50% Tails with probability 50% \100 The probability of actual toss to get head ??? Head Tail A coin does not remember bygones. To get the third head after getting head twice \100 \100 \100

  4. 1. The Law of Averages The Law of Averages (2) A run of heads just doesn’t make tails more likely next time. # of tosses # of heads # of tosses # of heads # of heads minus half the # of tosses # of heads minus half the # of tosses 10 4 -1 600 308 8 20 12 2 700 353 3 30 16 1 800 397 -3 40 20 0 900 458 8 50 25 0 1000 509 9 60 32 2 2000 1017 17 With a large number of tosses the size of difference between the number of heads and the expected number doesn’t decrease. 70 36 1 3000 1518 18 80 40 0 4000 2021 21 90 46 1 5000 2532 32 100 49 -1 6000 3030 30 200 100 0 7000 3524 24 300 159 9 8000 4018 18 400 205 5 9000 4521 21 500 255 5 10000 5055 55

  5. 1. The Law of Averages The Law of Averages (3) As the number of tosses increases, absolute size of probability error increases. Probability Error # of trials

  6. 1. The Law of Averages The Law of Averages (4) But when the number of tosses goes up, this percentage is goes down : the probability error gets smaller relative to the number of tosses. - The Law of Averages - Ralative percentage of probability error # of trials

  7. 2 3 The Stochastic Processes Making a Box Model INDEX The Law of Averages (Law of Large Numbers) 1

  8. \100 \100 \100 \100 100 100 100 100 2. The Stochastic Processes Stochastic Process and Applying a Box Model • To record the number of getting heads in a coin tossing • To see how much the house should expect to win at Roulette • To see how accurate unemployment rate through a sample survey is likely to be Able to analyze using stochastic process If you use a Box Model, you can comprehend comlicated stochastic processes.

  9. 2. The Stochastic Processes The Stochastic Processes and Box Model at Inference Inference : A process to find out something about the population using information from the sample The Stochastic Process A Box Model drawing tickets sampling box Population sample drawn tickets ditribution of tickets Population

  10. 1 2 3 The Stochastic Processes Making a Box Model INDEX The Law of Averages (Law of Large Numbers)

  11. 3. Making a Box Model Box of tickets drawing tickets • Pick one ticket at random. And make a note of the number on it. • Put it back in the box. And make a second draw at random again. • Having drawn twice at random with replacement, you add up the two numbers. 25 draws (with replacement) box 1 3 4 6 2 5 sum : 88 32462 35442 36412 41556 22255 Ten times repeated results 88 84 80 90 83 78 95 94 80 89 In principle, the sum could have been as small as 25 or as large as 150. But in fact, the ten observed values are all between 75 and 100.

  12. 1 2 3 4 5 6 1 3 4 6 2 5 25 draws 3. Making a Box Model Making a Box Model 1 What numbers go into the box? 2 How many of each kind? 3 How many draws?

  13. +1000 +1000 -1000 -1000 +1000 -1000 3. Making a Box Model Example A gambling problem in which the same bet is made several times 1 Which ticket? +1000 -1000 The tickets in the box show the amounts that can be won (+) or lost (-) on each play. 2 How many of each kind? The chance of drawing any particular value from the box equals the chance of winning that amount on a single play. 3 How many draws? 4 The number of draws equals the number of plays. Net gain? + …… +1000 -1000