Suppose the point-values were assigned to be 3 points, 2 points, and 1 point from the inside out for each dart board. Which dart board would you prefer to shoot at, assuming you want to win but your throw is random? Why?
**Extension 1:** What point values should be assigned to each location on each dart board to make that dart board mathematically fair? That is, the reward for getting a dart in a region and the area of the region are compensated for equally in the sense that it would not matter which region was hit...without changing the size of the respective dimensions.

**Extension 2:** Is it possible to not only make an individual dart board fair but also all three dartboards mathematically fair?

Source: Adapted from L. Winters' "Day in Life of J and K," *CMC Communicator*, Vol. 17#1

**Hint:** Start with a shape, such the square. What is the area of the three regions involved?

Plus, what does "mathematically fair" mean in this context?

**Solution Commentary:** A weighting is involved in determining probabilities (something called Monte Carlo!). Again, focus on the square. The areas of its three regions (inner to outer) are 1, 8, and 16. Thus, on a random toss, a dart will hit the inner region with probability of 1/25, the middle region with probability of 8/25, and the outer region with probability 16/25.

Now, procede with the other two shapes...and then collectively look at the data to decide which dart board is best...did this conclusion match your original visual intuition?