= = 1 2. (6) What is the probability that Jack will get a wooden bat and Peter will not = 25

Transcription

1 Homework 10, Math10170, Spring 201 Question 1 At Baseball practice, the coach gives each player on his squad a bat drawn randomly from the bag of bats which contains old wooden bats, 3 new aluminum Easton Redline bats and two Combat B1 s. Jack is first in line to receive a bat from the bag and Peter is second in line. Use a tree diagram to determine the probabilities in (a), (b) and (c) below. Draw Tree Diagram in the space below (a) What is the probability that both players will get a Combat B1 bat? = (b) What is the probability that Peter will get a wooden bat? = 4 90 = 1 2. (6) What is the probability that Jack will get a wooden bat and Peter will not =

2 Question 2 (The Curse of The Daiy Scoop) The sports section of a college newspaper, The Daily Scoop, features a player of the week each week. It is widely believed that the honor or being named player of the week brings with it a curse in the form of injury or a loss in a contest or a decrease in player or team performance in the weeks following the appearance. Ten percent of the player of the week features are devoted to prediction about performance in the near future. Unfortunately the predictions of the sports analysts for the newspaper are only 70% accurate. Of the rest of the athletes featured, twenty percent have consistently high performance (with a low standard deviation). For these athletes, the chances that they will experience a drop in individual performance in the weeks following their appearance is about 2% on average. Eighty percent of the athletes featured without predictions have a high standard deviation in performance statistics (less consistent) and on average, there is a 60% chance that they will experience a relatively large drop in performance in the weeks following their appearance. (a) Fill in the probabilities on the tree diagram below where the first step shows which category the featured player featured fits into, P (Prediction Made), C (No prediction made and consistent player) and LC (No prediction made and less consistent player). In the case where predictions are made, the second step shows whether (F) the player fulfills the predictions made or not (NF). In the cases where no predictions are made the second step shows whether the player has a decrease in performance stats (D) or nor not (ND) 2

3 (b) Regardless of team performance, illness injury and other misfortunes, what is the probability that a player featured in the player of the week section of The Daily Scoop will experience a misfortune in the form of a failure to fulfill the expectations of the sports analysts for the paper or a decrease in performance (if no predictions were made) in the weeks following their appearance. If M = Misfortune, then P (M) = (.1)(.3) + (.18)(.2) + (.72)(.6) = =.07. Question 3: Fourth Down, Go for it or kick it? In American Football, when a team gains possession of the ball, they have 4 chances to advance at least 10 yards. If the team does not advance at least 10 yards, they lose the ball after the fourth attempt. Instead of making a fourth attempt, many coaches either punt the ball down the field or kick for a field goal. Each play is called a down and the strategy of punting or kicking the ball on fourth down was challenged by David Roemer in his paper Do Firms Maximize? Evidence from Professional Football. It seems that coaches are hesitant to go for it on fourth down ( attempt a fourth down conversion) because of the criticism it draws. For example, such a play by Bill Belichick on Nov. 1, 2009 drew much criticism when it resulted in failure and the opposing team (The Colts) won the game with a touchdown. Brian Burke of AdvancedFootballStats.com has developed a calculator to calculate the winning probability for the team with the ball for any given state in the game. In this instance, we can see his estimated probabilities and a breakdown of his calculation of the probability of a win for the Patriots on the tree diagram below: With the score at in favor of the Patriots and two minutes left in the game, the Patriots attempted to gain the two yards necessary to get their fourth down conversion at their own 28 yard line. According to statistical evidence, Burke claims that a punt from this position gains 38 yards on average which would have placed the Colts on their own 34 yard line giving them about a 30% chance of making a touchdown. On the other hand the probability of success for a fourth down conversion was about 60% and if it failed, the chances that the opposing team would score a touchdown (from statistical evidence) from that position was about 3%. The chances of getting a field goal from their own 28 yard line was 0. With two minutes left in the game, the Colts had only one timeout left so a 3

4 successful conversion would have won the game for the Patriots and a touchdown would have (and did) won the game for the Colts. (a) Using the above diagram, calculate the probability (in the given situation) of a win for the offense using the strategy of punting the ball. P (Win with a Punt) = 0.7. (b) Using the above diagram, calculate the probability (in the given situation) of a win for the offense using the strategy of attempting a fourth down conversion. P (Win with an Att. Conv.) = (c) Which is the better strategy in this this situation? The better strategy is to attempt a conversion since it offers the highest chance of winning the game. 4

5 Question 4 In a game of squash between Player A and Player B, Player A is about to play the ball: Player A has two options, he can either play a drop shot or a lob. Player B can return with a drive or a boast. When Player B drives the ball after a drop shot by Player A, there is a 30% chance that Player A will go on to win the point. When Player B plays a boast after a drop shot by Player A there is 80% chance that Player A will go on to win the point. When Player B drives the ball after a lob by Player A, there is a 60% chance that Player A will go on to win the point. When Player B plays a boast after a lob by Player A there is 10% chance that Player A will go on to win the point. Use a tree diagram to find the optimal strategy for Player A, drop shot or lob assuming that Player B will counter with the play that maximizes his/her chances of winning the point. 0 Drop Lob Drive: P(A Wins) = 0.3 Boast : P(A Wins) = 0.8 Drive : P(A Wins) = 0.6 Boast: P(A Wins) = 0.1 Player A should play a Drop shot; If Player A plays a Lob, then Player B will play a Boast and A s chance of winning the point will be 0.1. On the other hand if Player A plays a Drop Shot, then Player B will counter with a drive and Player A s chance of winning the point will be 0.3.