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Averages(Dr. Monticino)

Assignment Sheet

Read Chapter 16Assignment #9 (Due April 6th)

Chapter 16 Exercise Set A: 1,2,3,4,5, 6,8 Exercise Set B: 1-7 Exercise C: 1,2 Review Exercises: 1-8,10

Overview

Law of AveragesAbsolute and relative error

Sum of DrawsBox Models

Roulette exampleTexas lottery

Law of Averages

The law of averages says that if a chance process is repeated a large number of times , then the percentage of times that a particular event occurs is likely to be close to the probability of that event Provided repetitions of the processes

are essentially identical and independent of one another

Law of Averages

If the experiment is repeated N times and the probability on each repetition that an event A occurs is p, then(#times A occurs)/N gets close to p as

N gets largeNote, that this not the same as saying

(#times A occurs) gets close to being exactly p*N

Examples A coin is tossed repeatedly. You win $100 if

exactly half the tosses are heads. Which is better: 2 tosses or 10 tosses 10 tosses or 100 tosses

Now you win $100 if the percentage of heads is between 40% and 60%. Which is better: 10 tosses or 100 tosses

2

1

2461.2

1

5

10 10

Examples

Suppose you play roulette in Nevada. Which is a better:Spin 50 times and win $1000 if get 40

or more redsSpin 100 times and win $1000 if get

80 or more reds

Sum of Draws

For a random process producing real number values, we are often interested in the sum of the numbers produced For example, if gambling, then your

net winnings/losses is an important quantity

Box Models

A box model is a useful way to represent a complicated chance process

Address the following when constructing a box modelWhich numbers go into the boxHow many of each numberHow many draws are being made

from the box

Examples

Construct a box model for playing “red-or-black” 10 times at a Nevada roulette table, making $1 stakes

Construct a box model for betting on “17” twenty times at a Nevada roulette table, making $1 stakes

Construct a box model for buying a ticket in the Texas lottery

(Dr. Monticino)