Part IV. Chance Variability – Chapters 16
Chapter 16. The Law of Averages
If you are tossing a coin and Heads comes up 4 times in a row, does the Law of Averages
say that a Tail is due? Does the coin change from toss to toss? If not, then how could the
chances of getting a H or T change. Does the coin know that you just got 4 Heads in a
row? Of course not. That is not what the Law of Averages says. What it says is given in
the list of key terms.
Key Terms
Law of Averages – Chance error is likely to be large in absolute terms, but small relative
to the number of tosses.
Box Model – The idea of drawing from a box which models a chance process in order to
analyze chance variability.
THE LAW OF AVERAGES
The Law of Averages can be explained through coin tossing. A fair coin will land
heads with a chance of 50%. When the coin is tossed many times, the number of heads
will be around half the number of tosses, but it will be off by some amount. This amount
is attributed to chance error.
number of heads = half the number of tosses + chance error
Chance error is likely to be large in absolute terms, but smaller relative to the number of
tosses. This is the Law of Averages.
For example, say a coin is tossed 1,000 times and there were 512 heads.
The expected number of heads is 500;
the chance error = 12 ;
the relative error=12/1000 times 100%=1.2%
Then say we toss a coin 10,000 times and the number of heads is 5,075.
This time the chance error is 75, which is significantly larger than 12.
This is what we mean by “as the number of tosses goes up, the chance error gets bigger in
absolute terms”.
The relative error is 75/10000 times 100%=0.75%.
This is much smaller than the relative error with 1000 tosses. Therefore, as the number of
tosses go up (1000 to 10000) the chance error gets larger in absolute terms (12 to 75), but
is smaller relative to the number of tosses (.012 to .0075).
The Law of Averages guarantees that the relative, or percent error, gets closer and closer
to zero as the number of tosses gets bigger. The absolute error does not have to do that.
Remember: With a large number of trials, you are guaranteed to be getting closer
and closer to the expected value.
You can take advantage of the Law of Averages if you are a betting person.
Example: You are betting $1 that you will get 60% or more heads. You have a choice of
either 10 tosses or 100 tosses. Which is better?
Definitely 10 tosses is better. Why? Because with more tosses you are almost certain to
get close to 50% heads, not 60% or more.
SUM OF THE DRAWS AND BOX MODELS
A box model is a conceptual aid for visualizing a chance process. For instance, in tossing
a coin we don’t actually need a coin which we will flip. We can view this as having 2
pieces of paper in a box. On one piece of paper we write a 1, indicating a Head, and on
the other we write a 0, indicating a Tail. Then tossing a coin and getting a Head is
EXACTLY the same as drawing the ticket with the 1 on it from the box. If we want to
toss the coin more than once, we simply put the ticket back in the box and draw again.
Thus the coin tossing experiment is represented as a box model.
There are three questions to answer when making a box model:
1. How many tickets are in the box?
2. What numbers are written on the tickets?
3. How many draws do we make from the box?
A Box Model for Betting on Red in Roulette:
A Nevada roulette wheel has 38 pockets numbered 1 through 36, 0 and 00. The pockets
numbered 1-36 alternate red or black while the 0 and 00 are both green. You can place a
bet that the ball will land in a red pocket. The ball is equally likely to land in any one of
the 38 pockets. Assume that a dollar bet was made on red.
To apply the box model we answer the 3 questions above.
1. How many tickets go into the box? 38, one for each pocket on the wheel, since
they are equally likely.
2. What numbers go on the tickets? Think of the red tickets as winners and the
others as losers. If you bet $1 and the ball lands in a red pocket, the rules say you
win your dollar back plus another $1. If the ball lands in any other color pocket
you lose your $1. So there is a +1 on 18 winning tickets and a –1 on 20 losing
tickets.
3. How many draws from the box? This is just the number of time we will play the
game.
At this point our box model has been created and it looks like the following:
18
20
+$1
–$1
With this box model we don’t need a roulette wheel anymore because the game is
EXACTLY the same as drawing a ticket out of the box at random, replacing it each time
for the next play.
Your NET WINNINGS= SUM of the Numbers on the Tickets on each draw.
That is, the sum of the n draws is how much you win (or lose if it is <0) when you play
the game n times.
In the next Chapter we study the SUM of DRAWS .
Example 1, Page 285, # 1
Solution
Option (iii): The Law of Averages says that as the number of draws increases, the chance
error gets bigger in absolute terms. This means that the number of 1’s is likely to be
different from 6,000. The Law of Averages also says that as the number of draws
increases the chance error gets smaller relative to the number of draws and in this case, it
will be small relative to the 10,000 draws.
Example 2, Page 286, #9
Solution
Option (ii): With more draws, the percentage of reds is likely to be closer to the
percentage in the box, which is something above 50%. When the percentage of reds
among the draws is above 50%, there are more reds than blues.