The Major League Baseball teams have a won-loss record over the last ten games.

Question

The Major League Baseball teams have a won-loss record over the last ten games. For example, a team may be 6-4 over the last ten games, meaning they have won 6 games and lost 4 games. Assume the Yankees had an 8-2 ten game run every day for a week (7 days.) Then
1. What is the best record they could have had for that week?
2. What is the worst?
3. Assuming they had an 8-2 ten game run for two weeks (14 days). What is the best record they could have had for that week? What is the worst?
4. Assuming they had an 8-2 ten game run for n consecutive days. Describe a way to determine their record for those n days.
5. How would this change if their record had been 6-4?
6. How would this change if their record had been k-(10-k)?
7. Find three functions which come up in the analysis of this problem.

Explanation / Answer

1.) The easiest way to approach this problem is to draw a picture. Use a 'w' for a win, and an 'l' for a loss. For example, the results of the first 10 games coming before the 7 days you care about may look like this: w w w w w w w w l l ... In order to keep an 8-2 record, the next day's result must be the same as the result that was 10 days before. So in the pattern above, the next 8 days must all be wins, followed by 2 losses. This pattern would repeat over and over again. In order to get the best record for the next week, you would want to replace as many of the next 7 days as possible with wins. Since the pattern above starts with 8 wins, the Yankees could go 7-0 over the next 7 days using that pattern. 2.) To find the worst record they could have over the next 7 days, you would want to choose a different starting pattern. Instead of having all of the losses at the end of the pattern, you would want to front-load the losses, so your pattern would look like the following: l l w w w w w w w w ... For the next 7 days, you would have 2 losses, then 5 wins, so their record would be 5-2. 3.) For 14 days instead of 7 days, you would want to start with the same patterns, but just continue to extend them out. So the best record the Yankees could have would be 12-2, and the worst record they could have would be 10-4. 4.) For n days, you would want to keep the two initial starting patterns, and just continue to extend them out. There would be 3 parts to calculate the records instead of drawing them out: a.) Divide n by 10 and round down to the nearest integer to get the number of full 10 game cycles. b.) Multiply 10 by the answer to part a, and then subtract that number from n to get the number of days remaining after the last complete cycle. c.) To find the number of wins, multiply 8 by the answer to part a. This will give you the number of wins from the full 10 game cycles. Then add the number of wins you would get from the days remaining after the last complete cycle you found in part b. If you wanted the best record possible, assume that the first 8 days of a cycle are wins, and the last 2 days of a cycle are losses. If you wanted the worst record possible, you should assume the first 2 days are losses, and the last 8 days are wins. 5.) If their record was 6-4 instead of 8-2, you would do the same thing, except assume the pattern for the best record had 6 wins followed by all 4 losses, and the pattern for the worst record had all 4 losses first, followed by the 6 wins. In part c above, you would just use 6 instead of 8 when finding the number of wins from the complete cycles, and use the two new patterns when finding the number of wins from the remaining days. 6.) If their record was k-(10-k), you would do the same thing again, except you would assume the pattern for the best record had k wins followed by (10-k) losses, and the pattern for the worst record had all (10-k) losses first, followed by the k wins. In part c above, you would just use k instead of 8 or 6 when finding the number of wins from the complete cycles, and use the two new patterns when finding the number of wins from the remaining days. 7.) I'll leave this up to you. There are several functions described in the parts above, so just pick the 3 you want.

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