10/10/11

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.

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.

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.

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.

Calculating Probabilities

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

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.

Calculating Probabilities

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

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.

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!

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.

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.

Calculating Probabilities

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