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Most Difficult Puzzle


3. No such q exists.
4. Chances for the Police in question 1 and for the Army in question 2 are the same (50 percent) regardless of the probability p. Police wins the tournament with 25 percent probability regardless of the probability q in question 3.

The Solution

Miners and Bankers are able to create some inequity in the chances to win the tournament solely due to the asymmetrical scoring system; a combination of a win and a loss is better than two ties. Why in this situation these two teams would tie is unknown. Perhaps in the real world it would never happen. However, it makes an interesting math puzzle. The actual solution could be quite long, so I will try to squeeze the main points within the limits of the puzzles page.

1. There are 243 different scoring combinations. They can be split based on the number of ties (excluding the tie between the Miners and the Bankers) as follows: If the number of ties is t, the number of possible combinations is or 5, inequities are not explored and the probability of making the final match for any team is 50 percent. However, if t = 2 or 3, then the Police team gets an advantage in just a few cases and the expectation of making finals increases to 42 2/3 (compared to 40) out of 80 (t=2) and 22 (compared to 20) out of 40 (t=3). The cases could be counted with some work and are left to the readers as an exercise. (Hint: Police will be at an advantage, for example, if one of the cheating teams ties all games.) Total probability of the Police team making the final game is

By setting to zero, you can find the answer.

2. This is very similar to the first case. Inequities arise when t = 3 or 4, where t is the total number of ties. Expectation of making finals decreases to 18²⁄³ (compared to 20) out of 40 (t=3) and 4 (compared to 5) out of 10 (t=4). Again, the cases are left as an exercise. (Hint: Army will be at a disadvantage if it ties all three games; this will be offset against the cases in which Police ties all three games.) Total probability of the Police team making the final game is

By setting to zero, you can find the answer.

3. Police wins the championship with 50 percent probability if it advances to the final game with certainty. The probability of the Police team’s making the finals equals . Since p is assumed to be uniformly distributed on [0, 1], the following must take place:1

By solving the integral equation, you find q > 1, which is impossible. A more interesting question: What probability q triggers the expectation of the Police advancing to the final game to be equal to one-half? The same logic applies as we need to solve the equation.0.
This equation solves for q = 8/13.

4. The old scoring system was fair in that it prevents inequities from occur- ring. Therefore, the probability of Army’s or Police’s winning the group stage is 50 percent, regardless of the Miners-Bankers strategy.