Here are some problems that require understanding probabilities and organizing your thoughts correctly before attempting to work them out.

1. You have a normal chessboard and a set of dominoes. The dominoes are just the right size to cover two squares on the chessboard. Assume you have enough dominoes to completely cover the chessboard. Now assume the center four squares are missing. Can you still cover the board with dominos and not have any empty spaces or overhanging half dominos? Now start with a whole board again the remove the two squares diagonally across from each other on the corners. Can you cover this board with no empty space and no overhanging dominos?

2. What is the lowest number of people in a room such that the probability of two of them having the same birthday is about 50%?

3. A constant-skill archer in competition shoots an arrow. He shoots another arrow. It is further out than the first. What is the probability his third shot will also be worse than the first one?

4. Two gamblers flip coins. One does 1,000 tosses. The other does only 100. Which gambler has the better chance of having equal number of heads and tails? Hint: consider the limits of the number of tosses each could make and have equal numbers of heads and tails in principle (i.e. more than one toss).

The answers will be forthcoming in a future article. If you can’t wait, e-mail me for a hint or the answer.

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.