Perhaps you will find the problem interesting (sent via. e-mail to my stude=
nts)
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D
Jan. 2, 1997 (Happy New Year)
OK students, here is a fascinating problem that can be solved with basic pr=
obability and a bit of simple algebra. <bold> I especially encourge AP STAT=
. students to look this over carefully ... and appreciate the power of the =
algebraic argument.</bold>
=20
You are in a tennis club. The club has a CHAMPION and another player who w=
e will call GOODY. Both CHAMPION and GOODY are better players than you, an=
d CHAMPION is better than GOODY. You are given an opportunity to play thr=
ee sets against these two players. The rules say you must alternate oppone=
nts. That is, you have two choices: You can play in one of the two follow=
ing sequences:
CHAMPION GOODY CHAMPION or GOODY CHAMPION GOODY
Now, let's assume you would win =241,000,000 if you could <underline>win tw=
o sets in a row</underline>. To liven this up, let's assume that CHAMPION =
and GOODY don't want you to win this money. OK, here's the question. <bold=
> Which of the two choices above would you make to maximize your chances of=
winning?</bold> That is, would you want to start of playing CHAMPION or s=
tart off playing GOODY, assuming that you must alternate players and win tw=
o sets games in a row (out of three sets) to collect =241,000,000?
The answer is shown below, but don't read below this line now if you want t=
o work on this interesting problem.
=3D=3D=3D
=3D=3D=3D
=3D=3D=3D
=3D=3D=3D
=3D=3D=3D
ANSWER (Solution):
Lets assume that x represents your probability of beating CHAMPION in a set=
and y represents your chance of beating GOODY in a set. Here are the cons=
traints:
0 << x << y << 1
Now, you can win two in a row in one of the following two ways
SET1 SET 2 SET 3
Win Win --- (third set doesn't matter here)
Lose Win Win
Here are the related probabilities that you will win (C=3DCHAMP,G=3D GOOD)
C G C G C =
G
WW x y y x
LWW (1-x) y x (1-y) x y
If you choose the order CGC, your probability of winning is
xy + (1-x)yx =3D xy(2-x)
If you choose the order GCG, your probability of winning is
yx + (1-y)xy =3D xy(2-y)
Now, since x << y, it follow that (2-x) > (2-y) and that
xy(2-x) > xy(2-y)
In other words, given the provided premise, you have a better chance if you=
choose to play CHAMPION first. This might seem counter-intuitive, since t=
his choice might have you playing the better player twice in a three set ma=
tch, but simple algebra, along with a bit of basic probability, dictates th=
e best choice here. =5BNote: In either sequence, you have to beat CHAMPIO=
N once. Given that CHAMPION is a better player, your probability of beatin=
g him/her once is better if you play two sets agains him/her rather than ju=
st one.=5D
Again, AP STAT. students...Appreciate the power of algebraic thinking that=
appears above.
DOC