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Expected value Law of Averages Central Limit Theorem

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Chapters 16, 17, and 18. Expected value Law of Averages Central Limit Theorem. Box Models. Flipping a coin n times, or rolling the same die n times, or spinning a roulette wheel n times, or drawing a card from a standard deck n times with replacement , …

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box models
BoxModels
  • Flipping a coin n times, orrolling the same die n times, or

spinning a roulette wheel n times, or drawing a card from a standard deck n times with replacement, …

  • Interested in the accumulation of a certain quantity? We can box model the process.
  • Actual outcomes are therefore abstractly represented by tickets.
how to make a box model
How to Make a Box Model
  • First draw a rectangular box.
  • Then write next to the box how many times you are drawing from it: n = …
  • What tickets go inside the box?
    • That depends on what valueyou could add on to a requested quantity each time you draw from the box!
    • Then, write the probability of drawing a particular ticket next to that ticket.
    • Examples: Chapter 16, #5-8
expected value
ExpectedValue
  • The expected value of n draws from the box is therefore given by:
    • EVn = n*EV1
  • The expected value of 1 draw from the box, also called the box average, is given by:
    • EV1= weighted average of tickets in box
        • = first ticket *probability of drawing first ticket + second ticket *probability of drawing second ticket + …
law of averages
Law of Averages
  • “The more you play a box-model-appropriate game, the more likely you get what you see of the box.”
  • “What is expected to happen will happen.”
  • Examples: Chapter 16, #1, #4
  • A consequence of the Law of Averages is that we should not hope to come away with a gain by playing many times – we will eventually come out as a loser if we play long enough.
standard error
Standard error
  • The expected value of n draws is given by EVn = n*EV1, where EV1 is the average of the box.
  • Now of course our actual accumulated total could differ somewhat from the expectation, and we call our typical deviation standard error, given by:
  • SEn = √n *SE1, where SE1 is the standard error of the box.
standard error of a box
Standard error of a box
  • Standard error of a box, or standard error of a single play, or standard error of a single draw, all mean the same thing.
  • For a box with only two kinds of tickets, valued at A and B respectively, and with probability of p and q of being drawn respectively, the standard error of the box is given by:
  • SE1=|A-B|* √(p*q)
  • Examples: Chapter 17 #10
continuity correction
ContinuityCorrection
  • This is related to the normal table we played with.
  • Now the EVn acts as the “Average”
  • And SEn acts as the “Standard Deviation”
  • Chapter 17, Question 3c
  • Continuity Correction is needed when you are dealing with discrete outcomes.
  • Suggestion: Draw the normal curve and label the average. Then judge where you want to be and in what direction you should shade; then standardize and look up percentages.
  • And so the new version of Standardization Formula:
c c and discrete outcomes
C.C. and Discrete Outcomes
  • When do we know we may use continuity correction?
  • That’s when the observed outcomes are discrete.
  • For example, if you are counting the number of democrats among a sample of 400 people, you can probably get 0, 1, 2, …, 399 ,or 400, but nothing else between any two numbers (such as 349.97)
  • Examples: All questions in Chapter 18 where the box model is a “COUNTING BOX” (box with only 0 and 1)
  • A non-example: Height of people in the US
  • Example: P(at least break even) = P(actual > -0.5)
  • Example: P(lose more than $10) = P(actual < -10.5)
  • Example: P(win more than $20) = P(actual >20.5)
  • Example: P(no more than 2300 heads) = P(actual < 2300.5)
central limit theorem clt
Central limit theorem (CLT)
  • The basis for what we did is called

Central Limit Theorem.

  • The Central Limit Theorem (CLT) states that if…
    • We play a game repeatedly
    • The individual plays are independent
    • The probability of winning is the same for each play
  • Then if we play enough, the distribution for the total number of times we win is approximately normal
    • Curve is centered on EVn
    • Spread measure is SEn
  • Also holds if we are counting money won
  • Note: CLT only applies to sums! See Chapter 18 Question 10.