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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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Chapters 16 17 and 18

Chapters 16, 17, and 18

Expected valueLaw of Averages Central Limit Theorem


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.


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