Materials: White paper 8.5 x 11'', ruler, penny and pencil.

In this exercise you will try to estimate the size of a penny indirectly, from random collisions and statistics alone!

1. Take a penny and trace its circumference on a 8.5 x 11" piece of paper approximately 15 - 20 times. Place circles (targets) randomly but keeping the distribution more or less uniform (not all of them in one corner).

2. Now cut approximately 200 small pieces of paper (bullets) with diameters about 0.1 - 0.2 (pieces can be squares) smaller than a penny (you can also use rice or any small seed). Sprinkle them evenly onto the paper (by sweeping your hand over the page as you drop them) from a height of 4 to 5 feet. This process simulates random collisions. Discard all bullets that miss the paper. Let N be the total number of bullets hitting the paper, and n be the number of bullets hitting the targets. The number N has to be large, at least 100.

3. The probability of a hit is equal to the total area occupied by targets divided by the total area A of the paper,

Probability of a hit = n/N = m*a/A ,

where m = # of pennies , and a = yet unknown area of a penny.

Knowing n, N, m, and A (use a ruler to measure A), estimate a indirectly as,

a (Monte Carlo)= nA/Nm

The name Monte Carlo means taking chance as a method of getting results (as in casino gambling).

4. To verify this result, measure directly the diameter D of the circle and calculate its area as,

a (direct measurement) =3.14*D^2/4

5. Compare the results above. How accurate is the indirect measurement of a ? The answer gets better for larger values of N . Discuss any sources of error.

Rutherford applied this brilliant idea in 1910 to determine the size of the atomic nucleus (which is too small to be measured directly) using alpha particles (Helium nuclei ) as bombarding particles.