Sanderson M. Smith
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SHORTER SERIES FAVORS WEAKER TEAMS (Copy of note sent to all of my students in October, 2002)
MATH POWER can demonstrate some interesting things.
If you are a baseball fan, you know that the Anaheim Angels just won a playoff series from the New York Yankees. Since many baseball fans consider the Yankees to be the best team in baseball, this result was a bit surprising.
In major league baseball, the first set of games in the playoffs is a 3 out of 5 situation. The first team to win three games wins the series. The next set of playoff games (and the World Series) is a 4 out of 7 situation. Many have argued against the "shortness" of the 3 out of 5 saying that it gives an inferior team a better chance of winning (as contrasted to playing 4 out of 7).
Can we demonstrate this with mathematical probability? You bet your Uncle Ukiah's ukulele that we can.
Assume that there are ten poker chips labeled 1,2,3,4,5,6,7,8,9,10 placed in a bag. You and I decide to play a game involving the chips. The chips are thoroughly mixed and one is randomly chosen. In a weak moment, I agree to the following conditions:
You win if the chosen chip 1,2,3,4,5,6.
I win if the chosen chip is 7,8,9,10.
Clearly, you are the better player in the sense that you have a 60% chance of winning a game and I have only a 40% chance. If you have MATH POWER, you would not want to bet on me unless you were provided odds that made it wise to do so.
Suppose that in a moment of weakness, you give me the choice of playing just one game OR playing a 2 out of 3 series with you.
What happens if I decide to play 2 out of 3? Well, if W represent a win for me, there are three ways I can win the series:
WW (win first two games, third game not necessary)
WLW (win first, lose second, win third)
LWW (lose first, win second, win third)
OK, the probability of WW is (.4)(.4) = 0.16 = 16%.
The probability of WLW is (.4)(.6)(.4) = 0.096 = 9.6%
The probability of LWW is (.4)(.4)(.6) = 9.6%.
Now, if you have MATH POWER, you realize that the probability I will win a 2 out of 3 series is
16% + 9.6% + 9.6% = 35.2%.
If I choose to play just a single game, my probability of winning is 40%.
What about a 3 out of 5 series?
The situation becomes a bit more complicated here, but MATH POWER can provide us with a meaningful response. Here are the ways I can win, along with the appropriate probabilities:
Win in 3 games
Probability = (0.4)3 = 0.064
Win in 4 games
Probability = 3(0.4)3(0.6) = 0.1152
Win in 5 games
Probability = 6(0.4)3(0.6)2 = 0.13824
The probability I would win a 3 out of 5 series is about 31.74%.
Now... what about a 4 out of 7 series?
Well, you can bet your Aunt Agatha's antique automobile that if I list all the possibilities (as was done in the 3 out of 5 case), things really get a bit detailed and unnecessarily complicated. Let's really use some MATH POWER to examine the 4 out of 7 situation.
Win in 4 games
Probability = (0.4)4 = 0.0256
Win in 5 games
OK, we must have
_ _ _ _ W
and we must fill the four blank slots with 3 W's and 1 L. We can do this in 4!/3! = 4 ways. In other words, there are 4 different ways I can in in 5 games.
Probability = 4(0.4)4(0.6) = 0.06144
Win in 6 games
We must have
_ _ _ _ _ W
and we must fill the five blank slots with 3 W's and 2 L's. We can do this in 5!/[(3!)(2!)] = 10 ways. There are 10 ways I can win in 6 games.
Probability = 10(0.4)4(0.6)2 = 0.09216
Win in 7 games
We must have
_ _ _ _ _ _ W
and we must fill the six blank slots with 3 W's and 3 L's. We can do this in 6!/[(3!)(3!)] = 20 ways.
Probability = 20(0.4)4(0.6)3 = 0.110592
The probability I would win a 4 out of 7 series is 28.98%.
In the Angels/Yankee baseball playoff series (year 2002), had the Anaheim Angels been given the choice of a 3 out of 5 OR 4 out of 7 with the New York Yankees, every ounce of MATH POWER would suggest that they should choose the 3 out of 5. (This, of course, is assuming that the Angels would concede the fact that the Yankees were a better team.)
In the National Football League Super Bowl, does the "best" team always win? This, of course, is a matter of opinion. The Super Bowl is only one game. Should an inferior team in the Super Bowl be glad that it is just one game and not a series of games?
MATH POWER TO ALL
"The pure and simple truth is rarely pure and never simple."
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