Transcript Chapter 16 Notes

Law
of
Averages
(Dr. Monticino)
Assignment Sheet
 Read Chapter 16
 Assignment #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 Averages
 Absolute and relative error
 Sum of Draws
 Box Models
 Roulette example
 Texas 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 large
 Note, 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
1
better:
 2 tosses or 10 tosses
 10 tosses or 100 tosses
2
10  1 
  
 5  2 
10
 .2461
 Now you win $100 if the percentage of heads
is between 40% and 60%. Which is better:
 10 tosses or 100 tosses
Examples
 Suppose you play roulette in Nevada.
Which is a better:
 Spin 50 times and win $1000 if get 40 or
more reds
 Spin 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 model
 Which numbers go into the box
 How many of each number
 How 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)