In this issue we look at how the Boston Red Sox missed the playoffs, in spite of a late-summer record that made at least a wild card appearance very likely. How improbable was their collapse?

1. 10/10/11

2. Calculating Probabilities
• At the end of August
the Boston Red Sox
had a two-game lead
over the Yankees and
a 9-game lead over
Tampa Bay.
• With 27 games left in
the season, Boston
looked good to make
the playoffs, at least as
a wildcard team.

3. Calculating Probabilities
• But by the end of
September, Boston
had won only 7 more
games and lost 20!
They missed the wild
card round by one
game.

4. Calculating Probabilities
• To see how improbable
this was, let’s go back
to the end of August to
“predict” the most likely
events over the next
month.
• We can treat the likely
outcomes using a
binomial distribution.

5. Calculating Probabilities
• We know we can
generate a binomial
distribution because:
– We know the probability
of success for each
successive game, using
Boston’s winning
percentage to that point.
– Each of the 27
remaining games is an
independent event.
– The outcome of each
game is either a win or a
loss.

6. Calculating Probabilities
• This is equivalent to
predicting the number
of successful coin flips
for a given number of
trials.

7. Calculating Probabilities
• To generate a binomial
probability distribution,
use a graphing calculator
with stats capability.
Generate two columns,
one for the number of
wins and one for the
probability of that
number of wins. Use the
BINOMPDF function on
the TI-Nspire.

8. Calculating Probabilities
• With Excel, use the
BINOMDIST function to
generate the probability
distribution function.

9. Calculating Probabilities
• Once you have created
the data, graph it.
• This is the probability
distribution function for
the Red Sox data. Notice
that the most likely
outcome is for the Sox to
win 14 to 18 games,
which would ensure that
they make the playoffs.

10. Calculating Probabilities
• The probability of winning
only 7 games is well
outside the most likely
outcomes.
• The calculated probability
of 7 wins is 0.0151%.
• How rare is this? The
probability of being struck
by lightning is 0.000357%.
Had the Sox only won 5
games, instead of 7, they
would have been in
lightning territory!

11. Calculating Probabilities
• But perhaps it’s more
accurate to use
Boston’s winning
percentage from
August, instead of the
cumulative percentage
for the whole season.
• Here is Boston’s
August record. We can
use this percentage
and rerun the binomial
distribution.

12. Calculating Probabilities
• The probability of 7 wins
in this scenario is roughly
three times higher
(0.0461%) but still well
outside the more likely
outcomes.

13. Calculating Probabilities
• The truth is, unlikely
events do sometimes
happen. They are so rare
that we marvel when
they occur.