### Tip of the Week #20
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*Why Flip a Coin: The Art and Science of Good Decisions*

by H. W. (Harold Warren) Lewis (1997, John Wiley & Sons, Inc., 206
p.)

This entertaining book is light reading about misconceptions about
probability. Twenty-three short chapters discuss brain-teasing problems ranging from
winning the office football pool to winning a battle. Some of the paradoxes confound the
intuition.

Although the book is sometimes short on substance, the examples provoke
thinking and should foster interest in decision making. Many of the examples relate to the
United States (certain laws and issues about voting) that may be of less interest to
readers in other countries. However, the relevance for other countries should be readily
apparent.

One of many interesting ideas is "Lanchester's Law," named
after English engineer Frederick Lanchester.

1. Assuming that your firepower is proportional to the number of your
units (accuracy, firing rate, and other characteristics being equal), the strength of your
forces is proportional to the *square* of your number of units.

2. The quantity that doesn't change as the result of engaging forces is
the *difference* between the *squares* of the number of units on each side.

For example, Army A has 5 units, and Army B has 3 units. The difference
of the squares of their units is 5^2 - 3^2 = 25 - 9 = 16. Lanchester's Law predicts the
outcome of their engagement as A ending with 4 units and B with none.

For fun, I wrote a QuickBASIC Monte Carlo simulation (LANCHAST.BAS) to
test Lanchester's Law. As with Lewis's example, one side (A) starts with 5 units, and the
other side (B) with 3 units. I assumed a .01 chance of a unit hitting a unit of the
opposing force on any particular volley. After 1,000 trials:
Average ending A force: 3.54 units
Average ending B force: 0.25 units
P(A wins) = .874
P(B wins) = .126
Average ending A when A wins: 4.05 units (apparently the 4 units predicted by Lanchester's
Law)
Average ending B when B wins: 1.99 units.
A simple simulation provides much more information for decision making
and can easily incorporate other details, such as comparative advantages, in the model. |

Watch for future tips that discuss other interesting examples from *Why
Flip a Coin?*

—John Schuyler, July 1997

Copyright © 1997 by John R. Schuyler. All rights reserved. Permission to copy with
reproduction of this notice.