# Crossroads in Mathematics

## Example 11

### Dart Board Problem (Geometry and Probability)

This problem combines the ideas of area and probability in a meaningful manner.

Problem. A circular dart board has a single dot right in the middle. For each dart that hits the board 5 points are awarded. If a dart lands closer to the center dot than the outer edge, an additional 10 points are awarded. If a dart is thrown at random so as to hit the board, what is the probability that the extra 10 points will be awarded?

This problem can be solved using simple geometry and illustrates that doubling all linear dimensions of a plane figure quadruples its area. Alternatively, a computer or calculator program based on a random number generator can be used to simulate the dart throwing, and the relative frequencies obtained can be used to estimate the theoretical probability. This so-called "Monte Carlo" simulation can be used to illustrate the relationships between the means of several trials obtained by individuals, groups, and the entire class--leading to a discussion of weighted means and the central limit theorem.

It is not being suggested that teachers should have students write the needed computer or calculator program. Teachers could supply students with the needed program, have interested students write the program, do a classroom demonstration, or have each student find a few distances without programming and then combine the individual frequencies.

Extension. If the dart board is square, the problem becomes more challenging. Using traditional methods, it would be a fairly difficult calculus problem, but it can be solved by the Monte Carlo method with no use of calculus (see Figure 3).

Figure 3. A simulation of 100 darts randomly hitting a square dart board.