Error Analysis in Experimental Physical Science

§3 - Backgammon 101

As mentioned in the previous section, the topic of error analysis requires some knowledge of statistics. Here we begin our very simple study.

Although we used the word "simple" in the previous sentence, perhaps surprisingly it was not until the sixteenth century that correct ideas about probability began to be formed

For example, an annuity is an investment in which a bank receives some amount of money from a customer and in return pays a fixed amount of money per year back to the customer. If a fairly young customer wants to buy an annuity from the bank, the probability is that he or she will live longer than an older customer would. Thus the bank will probably end up making more payments to a young customer than an older one. Note that the argument is statistical: it is possible for the young customer to get killed by a runaway bus just after buying the annuity so the bank doesn't have to make any payments.

Thus the probabilities say that if the bank wishes to make a profit, for younger customers it should either charge more for the annuity or pay back less per year than for older customers.

The lack of the concept of what today is called "simple statistics" prior to the sixteenth century meant, for example, that when banks in England began selling annuities, it never occurred to them that the price to the customer should be related to his/her age. This ignorance actually caused some banks to go bankrupt.

Similarly, although people have been gambling with dice and related apparatus at least as early as 3500 BCE, it was not until the mid-sixteenth century that Cardano discovered the statistics of dice that we will discuss below.

For an honest die with an honest roll, each of the six faces are equally likely to be facing up after the throw. For a pair of dice, then, there are 6 × 6 = 36 equally likely combinations.

Of these 36 combinations there is only one, 1-1 ("snake eyes"), whose sum is 2. Thus the probability of rolling a two with a pair of honest dice is 1/36 = 3%.

There are exactly two combinations, 1-2 and 2-1, whose sum is three. Thus the probability of rolling a three is 2/36 = 6%.

The following table summarises all of the possible combinations:

Probabilities for honest dice
Sum Combinations Number Probability
2 1-1 1 1/36=3%
3 1-2, 2-1 2 2/36=6%
4 1-3, 3-1, 2-2 3 3/36=8%
5 2-3, 3-2, 1-4, 4-1 4 4/36=11%
6 2-4, 4-2, 1-5, 5-1, 3-3 5 5/36=14%
7 3-4, 4-3, 2-5, 5-2, 1-6, 6-1 6 6/36=17%
8 3-5, 5-3, 2-6, 6-2, 4-4 5 5/36=14%
9 3-6, 6-3, 4-5, 5-4 4 4/36=11%
10 4-6, 6-4, 5-5 3 3/36=8%
11 5-6, 6-5 2 2/36=6%
12 6-6 1 1/36=3%

A histogram is a convenient way to display numerical results. You have probably seen histograms of grade distributions on a test. If we roll a pair of dice 36 times and the results exactly match the above theoretical prediction, then a histogram of those results would look like the following:

Histogram of the theoretical prediction for 36 rolls

Exercise 3.1: roll a pair of honest dice 36 times, recording each result. Make a histogram of the results. How do your results compare to the theoretical prediction?

It is extremely unlikely that the result of Exercise 3.1 matched the theory very well. This is, of course, because 36 repeated trials is not a very large number in this context. Clicking on the following button will open a new browser window that will give you an opportunity to explore what the phrase "large number" means. There you will also find some questions to be answered.

Exercise 3.2

One of the lessons that you may have learned from Exercise 3.2 is that even for a large number of repeated throws the result almost never exactly matches the theoretical prediction. Exercise 3.3 explores this further.

Exercise 3.3

After completing Exercise 3.3, you may wish to think about the fact that, except for the first, all of the combinations of number of rolls times number of repetitions gave a total of 36,000 total rolls. This means that if we had a single set of 36,000 rolls of a pair of dice we could get different width curves of the same shape just by partitioning the data in different ways.

Question 3.1. What is the probability of rolling a seven 10 times in a row?

Question 3.2. Amazingly, you have rolled a seven 9 times in a row. What is the probability that you will get a seven on the next roll?

backgammon game

Backgammon, like poker, is a game of skill disguised as a game of chance: the most skillful player wins in the long run. In backgammon the most skilled player is the one who best understands the material of this section. If both players are equally ignorant of this material, which one wins the game is, literally, a crap shoot.

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This document is Copyright © 2001, 2004 David M. Harrison

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