Chapters 16, 17, and 18

 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

, …  Interested in the accumulation of a certain quantity? We can box model the process.

 Actual outcomes are therefore abstractly represented by tickets .

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 on to a requested quantity each time you draw from the box!

value you could add  Then, write the probability of drawing a particular ticket next to that ticket.

 Examples: Chapter 16, #5-8

 The expected value of n draws from the box is therefore given by:  EV n = n*EV 1  The expected value of 1 draw from the box, also called the box average , is given by:  EV 1 = weighted average of tickets in box  = first ticket *probability of drawing first ticket + second ticket *probability of drawing second ticket + …

 

“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.

 The expected value of n draws is given by EV n = n*EV 1 , where EV 1 is the average of the box.

 total could differ somewhat from the expectation, and we call our typical deviation standard error, given by:  SE n = √n *SE 1 , where SE 1 standard error of the box.

is the

 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: SE 1 =|A-B|* √(p*q) Examples: Chapter 17 #10

     1.

2.

This is related to the normal table we played with. Now the EV And SE n n acts as the “Average” 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:

z



actual



EV

n

SE

n

         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)

     The basis for what we did is called Central Limit Theorem.

The Central Limit Theorem (CLT) states that  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

EV n

 Spread measure is

SE n

Also holds if we are counting money won Note: CLT only applies to sums! See Chapter 18 Question 10.