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1. Consider a series of contests between two teams (A and B) with the winner of the
series the first team to win three games. The results can be described by the following
tree diagram.
The letters describe the winners of each game.
The red circles indicate that
the series stops there because Team A has won three games.
The blue circles indicate
that the series stops there because Team B has won three games.
The letters are arrayed
in tiers which indicate how many games have been played.
For instance, all letters in the
second row from the bottom are results after four games.
The results of the series can also be put into a table like Table a1 or Table c1 below,
which takes each branch that ends in a circle and allots it to a section which belongs to
the winning team and the number of games in the series.
(The consecutive letters in a
branch description represent the winners of each game in the branch.
An asterisk in a
branch description means that the game was not played because one team had already
won three games.)
From such table, we can fill in a table like Table a2 or Table c2 below
which gives
, the number of games in the series.
The random variable
can only take
]]
on the values 3, 4, or 5.
a)
For this part, suppose that Team A wins each game with probability
(and, of course,
:
Team B wins with probability
).
Fill in the Table a1 which describes the results of
":
the series.
I have filled in two entries for you.
Now use Table a1 to fill in Table a2,
which contains the value and probabilities for the random variable
described above. I
]
have filled in all the probabilities in Table a2.
You must justify those values from the
values in Table a1 to get credit.
Finally, use the extra column in Table a2 to obtain the
expected length of the series,
.
IÐ] Ñ
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View Full Document Table a1:
Results Using Probability Model From Part a
Result of Series
Branch
Probability
Total Probability
A wins in 3 games
AAA**
AABA*
ABAA*
BAAA*
AABBA
ABABA
ABBAA
BAABA
BABAA
BBAAA
BBB**
BBAB*
A wins in 4 games
A wins in 5 games
B wins in 3 games
B wins in
:Ð":
Ñ
Ð" :Ñ
$#
$
4 games
B wins in 5 games
BABB*
ABBB*
BBAAB
BABAB
BAABB
ABBAB
ABABB
AABBB
Table a2:
Length of Series
C:
Ð
C
Ñ
C
:
Ð
C
Ñ
$"
$
:
$
:
%
$: *: "#: ':
&
': "#: ':
#
#$
%
%
Total
1
b)
Based on the expected value calculated in Part a, use calculus to show that the
maximum expected length of the series occurs for
(This should agree with your
: œ "Î#Þ
common sense which should tell you that it will take longer when the teams are evenly
matched.) Show that when
, the resulting maximum value of the expected length
: œ "Î#
of the series is then
%Þ
"
)
c)
We are now going to investigate what would happen if we tried to create a model
which would extend the expected length of the series.
Here is the model.
In each game,
each team puts in an effort to win which is proportional to one plus the number of games
the other team has won so far.
The probability of winning is then the effort of the given
team divided by the sum of the efforts of both teams.
In a formula, we have
:œ
Þ
A"
A A #
+,
,+
and
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This note was uploaded on 12/12/2010 for the course STAT 346 taught by Professor Staff during the Fall '08 term at George Mason.
 Fall '08
 Staff
 Probability

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