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 Angelsjust won a playoff series from theNew 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 5situation. The first team to win three games wins the series. The next set of playoff games (and the World Series) is a4 out of 7situation. Many have argued against the "shortness" of the3 out of 5saying that it gives an inferior team a better chance of winning (as contrasted to playing4 out of 7).Can we demonstrate this with mathematical probability? You bet your

Uncle Ukiah's ukulelethat 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 playerin 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 providedoddsthat made it wise to do so.Suppose that in a moment of weakness, you give me the choice of playing just

one gameOR playing a2 out of 3series 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 5series?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

WWW

Probability = (0.4)

^{3 }=0.064Win in 4 games

WWLW

WLWW

LWWWProbability = 3(0.4)

^{3}(0.6) =0.1152Win in 5 games

WWLLW

WLLWW

LLWWW

WLWLW

LWLWW

LWWLWProbability = 6(0.4)

^{3}(0.6)^{2}=0.13824

TOTAL PROBABILITY-->

0.31744The probability I would win a

3 out of 5series is about31.74%.Now... what about a

4 out of 7series?Well, you can bet your

Aunt Agatha's antique automobilethat if I list all the possibilities (as was done in the3 out of 5case), things really get a bit detailed and unnecessarily complicated. Let's really use some MATH POWER to examine the4 out of 7situation.

Win in 4 games

WWWW

Probability = (0.4)

^{4 }=0.0256Win 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) =0.06144Win 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}=0.09216Win 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

TOTAL PROBABILITY-->

0.289792The probability I would win a

4 out of 7series is28.98%.

===============================In the Angels/Yankee baseball playoff series (year 2002), had the Anaheim Angels been given the choice of a

3 out of 5OR4 out of 7with 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."

-OSCAR WILDE

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