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Ask The Wizard #380

You want to get from the bank to the casino by bus. There are two buses that make a loop stopping at both places continuously. It takes one bus 1/2 hour to complete a loop and the other bus 3/4 hour. You have no idea where they are in their loops. What is the average waiting time until the next bus?

anonymous

11 minutes, 40 seconds

Here is my solution (PDF).

The following puzzle is called Prime of Your Life. The object is to determine the relationship between the roll of five dice and it's score. The order of the dice does not matter. Here are some random rolls and their score:

  • Roll: 1,3,1,4,6 -- Points = 3
  • Roll: 4,4,3,3,3 -- Points = 9
  • Roll: 2,4,4,2,3 -- Points = 7
  • Roll: 4,5,1,1,5 -- Points = 10
  • Roll: 4,4,1,3,3 -- Points = 6
  • Roll: 5,1,4,1,2 -- Points = 7
  • Roll: 1,1,5,1,5 -- Points = 10
  • Roll: 5,6,4,5,6 -- Points = 10
  • Roll: 6,5,5,3,6 -- Points = 13
  • Roll: 4,2,6,1,2 -- Points = 4

Here are some other rolls. To verify you understand the formula for the points, how many points are in each roll?

  1. Roll: 1,5,4,2,2 -- Points = ?
  2. Roll: 3,4,3,5,5 -- Points = ?
  3. Roll: 2,6,6,1,3 -- Points = ?
  4. Roll: 3,5,3,3,2 -- Points = ?
  5. Roll: 4,3,4,1,5 -- Points = ?

anonymous

  1. 9
  2. 11
  3. 5
  4. 16
  5. 8

Now that March Madness 2023 is over, how has your probability of a perfect bracket and expected wins by seed changed?

anonymous

My strategy for a perfect bracket is to pick the higher seeded team (meaning a lower seed number) in every game. At the end, when it gets to 1 vs. 1 seed, pick randomly. Based on 38 seasons of data. Here are some probabilities for each possible game:

  • 1 vs. 16 seed = 99.31%
  • 2 vs. 15 seed = 93.06%
  • 3 vs. 14 seed = 84.72%
  • 4 vs. 13 seed = 78.47%
  • 5 vs. 12 seed = 64.58%
  • 6 vs. 11 seed = 62.50%
  • 7 vs. 10 seed = 60.42%
  • 8 vs. 9 seed = 51.39%

To survive the first round, you would need to win all eight of the games listed above four times each. The probability of that is 1/4354.

Here are the various matchups in round 2, assuming you make it that far using this strategy.

  • 1 vs. 8 seed = 80.00%
  • 2 vs. 7 seed = 69.51%
  • 3 vs. 6 seed = 60.00%
  • 4 vs. 5 seed = 56.00%

To survive the second round, you would need to win all four of these games listed above four times each. The probability of that is 1/811.

Here are the various matchups in round 3, assuming you make it that far using this strategy.

  • 1 vs. 4 seed = 73.44%
  • 2 vs. 3 seed = 60.71%

To survive the second round, you would need to win both of these games listed above four times each. The probability of that is 1/54.

In round 4 there will be 1 vs. 2 seed games. The probability of the 1 seed winning is 53.73%. The probability of that happening four times is 0.53734 = 0.112355.

In round 5, there will be two 1 vs. 1 games. The probability of winning each is 1/2, so the probability of winning two is 1/4.

In round 6, there will be one 1 vs. 1 games. The probability of winning is 1/2.

Taking the product of all these probabilities results in a probability of a perfect bracket of 1 in 13,569,150,522.

To answer your other questions, here are the average wins per team, by seed.

  • 1 seed = 3.29 average wins
  • 2 seed = 2.32 average wins
  • 3 seed = 1.85 average wins
  • 4 seed = 1.55 average wins
  • 5 seed = 1.16 average wins
  • 6 seed = 1.07 average wins
  • 7 seed = 0.9 average wins
  • 8 seed = 0.74 average wins
  • 9 seed = 0.59 average wins
  • 10 seed = 0.61 average wins
  • 11 seed = 0.63 average wins
  • 12 seed = 0.51 average wins
  • 13 seed = 0.25 average wins
  • 14 seed = 0.16 average wins
  • 15 seed = 0.11 average wins
  • 16 seed = 0.01 average wins