### Tip of the Week #20                     Tip Index

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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