Honor Code: You must do these problems on your own, and submit your answers no later than 11 PM on Sunday, October 14.
Name:
NetID:
(Note: This example takes place in an alternate - happier - universe.)
You've decided to supplement your income by selling "Go, Cubbies!" t-shirts outside the stadium before each of the home games the Cubs will play during the upcoming playoff series. On a clear day, you expect to be able to sell 2,000 shirts. However, if it rains, you expect to sell only 1,200.
On the morning of the first playoff game ("opening day") in Chicago, the National Weather Service forecasts a 40% chance of rain.
What is the expected number of shirts you'll sell on opening day? t-shirts
The playoff series will end when one team has won three games. The first two games will be played "at home" in Chicago, the next one (or two, if needed) will be played "on the road," and the final game - which will be played only if each team wins two of the first four - will be back in Chicago. After a bit of analysis of past data, you believe there's a 62% chance the series will end in 3 or 4 games (and a 38% chance the series will last 5 games). The long-range forecast calls for a 50% chance of rain on the day of Game 2, and a 20% chance of rain on the day tentatively scheduled for Game 5. (The weather varies independently from day to day.)
What is your expected sales total across the first two games? t-shirts
What is your expected sales total across the entire series? t-shirts
Just before the series begins, a scalper offers you a Game-5 ticket in Section 11 (this is a good seat) for $750. If Game 5 is played, you expect to be able to resell this ticket for $2100 (you'll be outside the stadium selling t-shirts, of course); however, if the series ends earlier than Game 5, you can only redeem the ticket for its $80 face value.
What is your expected profit if you buy this ticket? $
You believe that the Cubs have a 60% chance of winning each home game, and a 45% chance of winning each game on the road. (The outcome of each game is independent of the outcomes of the other games.)
What is the probability that the Cubs will win the series in 3 games? %
What is the probability that the series will last only three games? %
If the series lasts only three games (i.e., if the first three games are all won by the same team), how likely is it that the Cubs will be the victors? %
What is the probability that the Cubs will win the series? % (... and don't answer, "For the Cubs, it's always ZERO!")
Hint: This final problem can be solved analytically, but the computations are a bit tedious. If you wish, feel free to estimate the probability using 100,000 simulation runs.
When you are finished, click here: