2017/4/11 Section 5.7 (Part A) [43 points] WebAssign Section 5.7 (Part A) [43 points] (Homework) Current Score : - / 43 Huizhi Liu MA 114, section 601, Spring 2017 Instructor: Lavon Page Due : Tuesday, April 11 2017 11:00 PM EDT 1. -/6 points The Reds and the Cubs are playing 4 games. For each game, the probability that the Reds win is 0.58. The probability of the Reds winning is not affected by who has won any previous games. (a) What is the expected value for the number of games the Reds will win? (Give your answer correct to two decimal places.) (b) What is the expected value for the number of games the Cubs will win? (Give your answer correct to two decimal places.) 2. -/7 points A baseball player has a .255 batting average. In other words, he gets a hit with probability 0.255 each time he comes to bat. (a) If he comes to bat 4 times in a game, what is the probability he gets exactly 3 hits? (b) If he comes to bat 4 times in a game, what is the expected value for the number of hits he will get? 3. -/9 points A certain baseball player hits a home run on the average of once every 8 times at bat. Use the idea of independent trials to answer the following questions. (a) If he bats 5 times in a game, what is the probability he will hit exactly one home run? (b) If he comes to bat 5 times, what is the probability he will hit at least one home run? (c) If he comes to bat 5 times in a game, what is the expected value for the number of home runs he will hit? http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349090 1/2 2017/4/11 Section 5.7 (Part A) [43 points] 4. -/9 points Data shows that 93% of the people in a certain population are righthanded. A group of 10 people from this population are selected at random. (a) What would be the expected value for the number of righthanded people in the group? (b) What is the probability that exactly 8 of the people in the group are righthanded? (c) What is the probability that at least 8 of the people in the group are righthanded? 5. -/12 points Faye is taking 4 courses this semester. She thinks that in each course she has a 90% chance of passing, and she thinks that her chances of passing each course do not depend on whether she passes any of the other courses. [This means that it is valid to think of the 4 courses as "independent trials" where in each course the possible outcomes are "pass" and "fail" and in every case the probability of passing is 0.9 and the probability of failing is 0.1.] (a) What is the probability that she passes all 4 of her courses? (b) What is the probability she passes exactly 3 of the courses? (c) One of the 4 courses is English. What is the probability that she fails English and passes the other 3 courses? (Hint: This is not the same question as part (b). In part (b) we didn't care which of the 4 classes she failed. Here we want to force the failing course to be English.) (d) What is the expected value for the number of courses she will pass? http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349090 2/2 2017/4/11 Section 5.7 (Part B) [38 points] WebAssign Section 5.7 (Part B) [38 points] (Homework) Current Score : - / 38 Huizhi Liu MA 114, section 601, Spring 2017 Instructor: Lavon Page Due : Tuesday, April 11 2017 11:00 PM EDT 1. -/6 points Kim and Susan are playing a tennis match where the winner must win 2 sets in order to win the match. For each set, the probability that Kim wins the set is 0.59. The probability of Kim winning the set is not affected by who has won any previous sets. Draw a tree to represent this tennis match and answer the following questions. (a) What is the expected value for the number of sets that will be played? (Hint: We only care about the number of sets played, not who wins.) (b) What is the expected value for the number of sets that Kim will win? (Hint: We only care about the number of sets Kim wins during the match, not if she ended up winning the match in the end.) 2. -/9 points Edna is trying to pass a competency exam. Each time she takes the exam she has a 40% chance of passing. She is allowed a maximum of two attempts. Draw a tree diagram to represent her attempts to pass the exam and answer the following questions. (a) How many outcomes does your tree show? (b) What is the probability she will eventually pass the exam? (c) What is the probability she will take the exam exactly twice? (d) What is the expected value for the number of times she will take the exam? (e) What is the expected value for the number of times she will fail the exam? http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349091 1/4 2017/4/11 Section 5.7 (Part B) [38 points] 3. -/9 points A student is trying to pass a competency exam. Each time she takes the exam she has a 35% chance of passing. She is allowed a maximum of three attempts. Draw a tree diagram to represent her attempts to pass the exam and answer the following questions. (a) How many outcomes does your tree show? (b) What is the probability she will eventually pass the exam? (c) What is the probability she will take the exam exactly three times? (d) What is the expected number of times she will take the exam? (e) What is the expected number of times she will fail the exam? 4. -/5 points Roulette is one of the most common games played in gambling casinos in Las Vegas and elsewhere. An American roulette wheel has slots marked with the numbers from 1 to 36 as well as 0 and 00 (the latter is called "double zero"). Half of the slots marked 1 to 36 are colored red and the other half are black. (The 0 and 00 are colored green.) With each spin of the wheel, the ball lands in one of these 38 slots. One of the many possible roulette bets is to bet on the color of the slot that the ball will land on (red or black). If a player bets on red, he wins if the outcome is one of the 18 red outcomes, and he loses if the outcome is one of the 18 black outcomes or is 0 or 00. So, when betting on red, there are 18 outcomes in which the player wins and 20 outcomes in which the player loses. Therefore, when betting on red, the probability of winning is 18/38 and the probability of losing is 20/38. When betting on red, the payout for a win is "1 to 1". This means that the player gets their original bet back PLUS and additional amount equaling their bet. In other words, they double their money. (Note: If the player loses they lose whatever amount of money they bet.) Scenario: Mike goes to the casino with $300 in his pocket and decides to place a single $50 bet on red. He knows that if he loses he will have $50 less than he started with. However, if he wins he doubles his $50, puts it in his pocket with the rest of his money, and walks away. What is the expected value for the total amount of money he will have in his pocket after he has made this bet (and either won or lost)? (Give your answer correct to the nearest cent.) $ http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349091 2/4 2017/4/11 Section 5.7 (Part B) [38 points] 5. -/9 points Challenge Problem! This problem will put your understanding of the concepts learned in this section to the test. Detailed hints and tips are provided to help you along, but this problem will still take a good amount of time and effort on your part. Be patient and don't be discouraged if you don't "get it" right away. This problem is designed to challenge you are you up for the challenge? In basketball, a oneandone free throw situation sometimes occurs when a player is fouled. The fouled player gets to take a shot from the freethrow line, and if he makes the first shot then he gets a second shot. If he misses the first shot, that's the end and no points are scored. Each basket made is worth 1 point. So when it's all over, he will have scored 0, 1, or 2 points. Scenario 1: Ann is a 72% free throw shooter. Let's assume that her free throw attempts are independent trials, i.e. every time she shoots there will always be a 0.72 probability that she makes the shot. Hints and Tips: Draw a tree diagram of this situation, put in the numbers, and "multiply up" the branches to find the probabilities of each of the three outcomes. Use this tree and the concept of expected value to answer the following question. What is the expected value for the number of points Ann will score if she is fouled in a oneandone situation? (Give your answer correct to three decimal places.) Scenario 2: Ahmed is a basketball player who averages scoring 1.6 points whenever he has a one andone free throw situation. In other words, the expected value for the number of points Ahmed will score if he is fouled in a oneandone situation is 1.6. Use this information to calculate what Ahmed's freethrow shooting percentage is. Hints and Tips: We'll use the symbols "p" and "E" to represent what we're talking about. Let: p = success probability whenever Ahmed shoots a free throw E = expected value for number of points scored in a oneandone situation In Ann's case, we were given that p = 0.72, and we used that information to draw a tree diagram and solve for E. Conversely, in Ahmed's case we know that E = 1.6, and we want to find the value of p. To do this, use exactly the same procedure that you used in Scenario 1, except instead of working with the number 0.72, simply use the symbol p in your tree diagram. Use this tree to write an equation involving p and E. Plug in the value of E that you were given for Ahmed into the equation and solve for p. (Recall from previous math courses that to solve an equation of this nature you'll need to use the quadratic formula. If you've forgotten the formula simply do a Google search for "quadratic equation solver" there are lots of good, free ones out there for you to use. You have to get the equation in the right format, but once you do that you can use a solver to do all calculations.) You'll find that there are actually two solutions for p. However, in this case p is a probability and we know that probabilities must be values between 0 and 1 so only one of the values is a valid answer. http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349091 3/4 2017/4/11 Section 5.7 (Part B) [38 points] What is Ahmed's probability of success when shooting free throws? (Give your answer correct to three decimal places.) http://www.webassign.net/web/Student/AssignmentResponses/last?dep=15349091 4/4 For ans1. 4.(a) What would be the expected value for the number of right-handed people in the group? 0.93 Incorrect: Your answer is incorrect. For ans2. 1(b) What is the expected value for the number of sets that Kim will win? (Hint: We only care about the number of sets Kim wins during the match, not if she ended up winning the match in the end.) 0.8946 Incorrect: Your answer is incorrect. 2. (c) What is the incorrect probability she will take the exam exactly twice? 0.24 (e) What is the expected value for the number of times she will fail the exam? incorrect 1.32 3. (d) What is the expected number of times she will take the exam? incorrect 2.9175 (e) What 2.3189 is the expected incorrect number of times she will fail the exam