World Series Analytical Solution


(if the probability NL wins an individual game = 1/2)

# of games to win the seriesexplanationprobability

1

The series must take more than 1 game, because the champion must win 4 games.
0

2

The series must take more than 2 games, because the champion must win 4 games.
0

3

The series must take more than 3 game, because the champion must win 4 games.
0

4

In order for the series to end after four games, one team must win the first four games in a row.
(1/2)(1/2)(1/2)(1/2) + (1/2)(1/2)(1/2)(1/2) = 2/16 = 1/8 = 0.1250

5

In order for the series to end in 5 games one team must win exactly 3 out of the first 4 games (in any order) and then win the fifth game.
C(4,3) (1/2)(1/2)(1/2)(1/2)(1/2) + C(4,3) (1/2)(1/2)(1/2)(1/2)(1/2) = 8/32 = 1/4 = 0.2500

6

In order for the series to end in 6 games one team must win exactly 3 out of the first 5 games (in any order) and then win the sixth game.
C(5,3) (1/2)(1/2)(1/2)(1/2)(1/2)(1/2) + C(5,3) (1/2)(1/2)(1/2)(1/2)(1/2)(1/2) = 20/64 = 5/16 = 0.3125

7

In order for the series to end in 7 games one team must win exactly 3 out of the first 6 games (in any order) and then win the seventh game.
C(6,3) (1/2)(1/2)(1/2)(1/2)(1/2)(1/2)(1/2) + C(6,3) (1/2)(1/2)(1/2)(1/2)(1/2)(1/2)(1/2) = 40/128 = 5/16 = 0.3125

Now check these answers. Remember, the total probability of all possibilities must equal 1.

Total probability = 0 + 0 + 0 + 0.1250 + 0.2500 + 0.3125 + 0.3125 = 1

As you can see, these numbers do check out. Now we find the expected value, E(x), by multiplying each value by its respective probability and adding them all together:

E(x) = 1(0) + 2(0) + 3(0) + 4(0.1250) + 5(0.2500) + 6(0.3125) + 7(0.3125) =

0 + 0 + 0 + 0.5000 + 1.2500 + 1.8750 + 2.1875 = 5.8125

Thus, by averaging all the trials from your model, you should get something near 5.8125 as your answer.

Questions:

  1. Should the average of the trials from your model be exactly 5.8125?

  2. Does an expected value of 5.8125 mean that we expect someone to win 0.8125 of a game?

  3. How accurate are the answers you got from your models?

  4. What does this expected value mean?

If you like, you may return to the original problem.

Please send questions and comments to Jay Hill