On September 7, 2008, the Pittsburgh Pirates lost their 82nd game of the 2008 season and tied the 1933–1948 Philadelphia Phillies major sport record (baseball, football, basketball, and hockey) for most consecutive losing seasons at 16. One year later on September 7, 2009, they lost their 82nd game of the 2009 season, and the record became theirs alone. The only way things could get much worse for the Pirates was to lose their 82nd game earlier in the season. Sure enough, on August 21, 2010, they lost their 82nd game of the 2010 season, extending their streak to 18 consecutive seasons. A major league baseball season consists of 162 games, so for the Pirates to end their streak, they will eventually need to win at least 81 games in a season.
a. Over the course of the streak, the Pirates have won approximately 42% of their games. For simplicity, assume the number of games they win in a given season follows a binomial distribution with n = 162 and p = 0.42. What is their expected number of wins in a season?
b. What is the probability that the Pirates will win at least 81 games in a given season? (You may use technology to find the exact binomial probability or use the normal distribution to approximate the probability by finding a z-score for 81 and then evaluating the appropriate area under the normal curve.)
c. Can you think of any factors that might make the binomial distribution an inappropriate model for the number of games won in a season?