Stat 104 Section Notes 4
Problem Solutions
1) In a 7 game playoff series like the World Series, the first team to win 4 games wins the
series.
a) Assuming there is equal probability that each team wins each game, what is the
probability that the series ends in 4, 5, 6, or 7 games?
Let X = number of games the World Series will go this year.
Let there be two teams in
the series: the Phillies (P) and the Red Sox (R).
Then, for the series to end in four
games with the Phillies as champions, the only way this can happen is if the Phillies
win the first four games, so the sequence of gamebygame winners would be: P
P
P
P
.
In each of those four games, the Phillies have 0.5 chance of winning each game, so the
probability that the Phillies win in 4 games is (0.5)
4
.
The same argument could be
made for the Red Sox winning in 4 games (and thus we just multiply the (0.5)
4
times
two):
.
For the series to end in five games with the Phillies as champions, there are 4 ways
(gamebygame sequences) this can happen: 1) R
P
P
P
P
, 2) P
R
P
P
P
, 3)
P
P
R
P
P
,
4) P
P
P
R
P
[Note, the Phillies have to win the 5
th
game].
This is equivalent to
choosing which 3 of the first 4 games the Phillies must win, thus there are
ways to do
this.
Also in each of those five games, the Phillies have 0.5 chance of winning or
losing each game, so the probability that the Phillies win in 5 games is 4*(0.5)
4
.
The
same argument could be made for the Red Sox winning in 5 games (and thus we just
multiply the 4*(0.5)
4
times two):
.
Similar arguments can be made to calculate the remaining two possibilities as such:
or
Note: coming up with these calculations is difficult, and would be too challenging for a
midterm question (though you would be expected to do them in a HW problem since
you would have more time to think).
b) Let
X
be the random variable for the number of games the World Series will go this
year.
What is the mean and variance of
X
?
Since X is a discrete random variable, we can use the following formulas:
Mean:
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View Full DocumentVariance:
Thus, the standard deviation is:
Since the World Series went to 7 games (in 1923), the number of games played has been:
4 games
5 games
6 games
7 games
Frequenc
y
17
18
18
33
c) Anything interesting seen above?
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 Fall '11
 MichaelParzen
 Probability

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