Colorado Rockies: Winning Streaks and Losing Streaks

As we all know by now, our beloved Colorado Rockies are a team that is prone to streaky play. In other words, when they are winning, it seems like they will never lose again. On the other hand, there are times when they are losing that it seems like they will never win another game! Being the statistical sport that baseball is, Joe Scoreboard has made the following observations concerning the Rockies. That is, when the Rockies win a game, there is a 70% chance that they will win their next game. However, when the Rockies lose a game, they have only a 40% chance of winning their next game. Use this data to answer the following questions.

1. Write a transition matrix describing the situation above.
T (2x2 Matrix) What must the sum of each row equal ?

2. Suppose that the Rockies won their first game of the season. What would be the initial distribution (initial state) matrix ?

D_0 (1x2 matrix) What does each element represent ?

3. Given the conditions from 1 and 2, find the Rockies' probable winning percentage after 1, 2, 3, 4, 5, and 10 games.

For example, D_10 = D_0 T^10 What does D_10 represent ?

Having played 10 games, could the Rockies actually have a record represented by this probability ? Why or why not ?

4. Now, recalculate example 3 with the condition that the Rockies lost their first game. Complete the chart below and compare the results.

Winning percentage having:


  Winning Percentage having
  won first game lost first game
1 game ____________ ____________
2 games ____________ ____________
2 games ____________ ____________
3 games ____________ ____________
4 games ____________ ____________
5 games ____________ ____________
10 games ____________ ____________


5. Notice that there is very little difference between the results after 5 games and after 10 games. What do these percentages represent ?

6. Are there any absorbing states in this example ?

7. The Rockies play a 162 game regular season. Predict their final won - lossrecord assuming that they win their first game and also if they lose their first game. How do the records compare? In this example, how important is the initial state ?

The Discrete Mathematics Project