In the classic pistol duel, the two duelists start back to back. On command they march a prescribed number of paces away from each other, and then turn to face each other, pistols at their sides. Then they raise their smooth bore pistols and each fire a single bullet at the other. If one duelist hits the other, he is considered the winner, while if both are hit or both are missed the duel is considered a draw. It seems to make sense that firing your pistol quickly might be advantageous as you could hit your combatant before he fires and hence throw off his shot. On the other hand, if you wait longer (and presumably aim more accurately) you stand a better chance of hitting your opponent. If you wait long enough to see that your opponent fired his weapon, and you are still standing, you could take as long as you wanted to line up your shot.

We will simulate two simplified versions of this dual. The first will be a “noisy duel” where each duelist can tell when his opponent fires his weapon and the second a “silent duel” where this information is not available. For both of these games we will make the following assumptions:

• The duelists must fire their weapons after 1, 2, 3, 4, 5, or 6 seconds.

• If a duelist fires after second i, the probability of hitting their opponent is i/6.

• In the noisy duel, if the duelists do not fire their guns at the same time, the duelist who fires second will wait until after 6 seconds thus guaranteeing that he hits his opponent.

Game 1: The Noisy Duel

To simulate this game we need two duelists with a piece of paper and a pair of dice. Before the game starts, each duelist writes down a number from 1 to 6 representing when he plans to fire his weapon. The two numbers are then compared. If the numbers are not equal, the player with the lower number will roll his die to simulate firing his weapon. If his number is i and he rolls 1, 2, …,i then he hits his target and wins the duel. If he misses, the other player will win as he will wait until after 6 seconds to fire and will be guaranteed a hit. If both players picked the same number, they both roll their die to simulate firing their weapon and determine the outcome in the same manner.

Activity 1: Below are some strategies represented as ordered pairs. The pair (i ,j) represents Player 1 planning to fire after i seconds and Player 2 planning to fire after j seconds. Below each ordered pair is one or two die rolls. In each case, determine the outcome of the duel.

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