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Colorado Technical University MATH 451 Math 451 estimating probability values Learning Materials You will visit an original scenario and then consider how a survey company provided them the resul... ts of the survey. Consider a company’s outcome for four different decision-making options. The following table shows the payoff of each alternative. In this example, you are considering using a survey company that offers some sample information which may not be very accurate for better decision making for a value of $5,000. They have provided the following probabilities for all alternatives you consider here: The survey company revealed the following information regarding the tips they were going to provide the company making better decision regarding their expansion. They conducted 40, 25, and 30 surveys regarding high demand, moderate demand, and low demand, respectively. The result and the probabilities of each case are shown in Figure 1. Figure 1: Results and the Probabilities of Each Case Calculate the Posterior Probabilities From Figure 1, you can calculate the posterior probabilities as follows: The rest of the calculations for probability revision given a positive survey results are tabulated as follows: Figure 2: Probabilities Revisions of the Survey for Positive Responses (PS) The calculations for probability revision given a negative survey results are tabulated as follows: Figure 3: Probabilities Revisions of the Survey for Negative Responses A company is considering using a survey company that offers some sample information which may not be very accurate for better decision making for a value of $3,000 in this example. They have provided the following data for all alternatives that are consider here. Question 1 What is the revised probability of having the low demand (LD) for a negative response (NS)? P(LD|NS) = 0.8715 P(LD|NS) = 0.0340 P(LD|NS) = 0.3715 P(LD|NS) = 87.5% You work for the spy agency CONTROL, and their arch enemy is KAOS, which is incorporated in Delaware for tax reasons. Recruiting agents is hard because not many people are qualified. Only about 1 in 397 people are qualified. There is a screening test. About 95% of the time, a qualified individual will pass the test, and about 3% of the time, an unqualified individual will get lucky and also pass the test. If a random person is chosen to take the test and he or she passes, what is the probability that the person is qualified? Round your answer off to the nearest tenth of a percent. 90.5% 10.8% 7.4% 89.2% Suppose that you have 2 bags of marbles, creatively labeled bag 1 and bag 2. Each bag contains a mix of green and yellow marbles in proportion 1/3 to 2/3, with one bag containing 1/3 green and the other containing 1/3 yellow. A sample with replacement from bag 1 yields 4 green marbles and 1 yellow marble. A sample with replacement from bag 2 yields 8 yellow marbles and 4 green ones. What are the probabilities that bag 1 has a majority of green marbles and that bag 2 has a majority of yellow marbles? Express your answers as percentages rounded off to the nearest tenth. Hint: You can use =BINOM.DIST() in Excel to compute the probabilities of the draws from the bags. 88.9%, 94.1% 85.2%, 86.6% 86.6%, 94.1% 85.2%, 93.3% A company is considering using a survey company that offers some sample information which may not be very accurate for better decision making for a value of $3,000 in this example. They have provided the following data for all alternatives that are consider here. Question 4 What is the probability of having a positive response (PS) for the median demand (MD)? P(PS|MD) = 0.125 P(PS|MD) = 0.84 P(PS|MD) = 60% P(PS|MD) = 40% A company is considering using a survey company that offers some sample information which may not be very accurate for better decision making for a value of $3,000 in this example. They have provided the following data for all alternatives that are consider here. Question 1 What is the revised probability of having the low demand (LD) for a positive response (PS)? P(LD|PS) = 12.5% P(LD|PS) = 0.7940 P(LD|PS) = 0.1588 P(LD|PS) = 0.0566 A company is considering using a survey company that offers some sample information which may not be very accurate for better decision making for a value of $3,000 in this example. They have provided the following data for all alternatives that are consider here. Question 2 What is the revised probability of having the median demand (MD) for a negative response (NS)? P(MD|NS) = 40% P(MD|NS) = 0.3715 P(MD|NS) = 0.0340 P(MD|NS) = 0.2380 A company is considering using a survey company that offers some sample information which may not be very accurate for better decision making for a value of $3,000 in this example. They have provided the following data for all alternatives that are consider here. Question 3 What is the probability of having a negative response (NS) for the low demand (LD)? P(NS|LD) = 12.5% P(NS|LD) = 0.875 P(NS|LD) = 0.16 P(NS|LD) = 0.125 Suppose that your friend has 3 coins, 2 loaded and one fair. Coin A comes up heads 80% of the time, coin B is fair, and coin C comes up tails 70% of the time. Your friend pulls a coin at random out of his pocket and offers you a bet where he gets $1 if the heads side appears. You are a bit uncomfortable with this and insist on testing the coin. Without knowing any more, you assess the probability of getting any one of the coins as 1/3. Your friend lets you test the coin by flipping it once, and it comes up heads. Given this new information, what is the probability that you have coin A (i.e., the one loaded for heads)? 0.667 0.467 0.767 0.500 Suppose that your friend has a loaded coin that comes up heads 80% of the time and another coin that looks just like the weighted one but is fair. Your friend pulls a coin out of his pocket and offers you a bet where he gets $1 if the heads side appears. Your estimate of the probability that he has pulled the loaded coin is 90%. You are a bit uncomfortable with this, and insist on testing the coin. You flip it 5 times, and the heads side comes up once. Given this new information, how likely is it that you have the fair coin? Express your answer as a percentage rounded off to the nearest whole number. Hint: You can use =BINOM.DIST() in Excel to compute the probabilities of the coin flips. 61% 60% 73% 97% Question 6 Suppose that your friend has a loaded coin that comes up heads 70% of the time and another coin that looks just like the weighted one but is fair. Your friend pulls a coin at random out of his pocket and offers you a bet where he gets $1 if the heads side appears. You are a bit uncomfortable with this and insist on testing the coin. You flip it 5 times, and heads come up 3 times. Given this new information, how likely is it that you have the loaded coin? Round your answer off to the nearest thousandth. Hint: You can use =BINOM.DIST() in Excel to compute the probabilities of the coin flips. 0.362 0.743 0.497 0.912 Suppose that your friend has 3 coins, 2 loaded and one fair. Coin A comes up heads 70% of the time, coin B is fair, and coin C comes up tails 70% of the time. Your friend pulls a coin at random out of his pocket and offers you a bet where he gets $1 if the heads side appears. You are a bit uncomfortable with this and insist on testing the coin. Without knowing any more, you assess the probability of getting any one of the coins as 1/3. Your friend lets you test the coin by flipping it once, and it comes up heads. Given this new information, what probability do you assign each coin? Round your answer off to the nearest thousandth. A 0.467 A 0.667 A 0.527 C 0.250 Suppose that your friend has a loaded coin that comes up heads 70% of the time and another coin that looks just like the weighted one but is fair. Your friend pulls a coin at random out of his pocket and offers you a bet where he gets $1 if the heads side appears. You are a bit uncomfortable with this and insist on testing the coin. You flip it 10 times, and heads come up 4 times. Given this new information, how likely is it that you have the fair coin? Round your answer off to the nearest thousandth. Hint: You can use =BINOM.DIST() in Excel to compute the probabilities of the coin flips. 0.759 0.232 0.848 0.302 A company is considering using a survey company that offers some sample information which may not be very accurate for better decision making for a value of $3,000 in this example. They have provided the following data for all alternatives that are consider here. Question 3 What is the revised probability of having the low demand (LD) for a positive response (PS)? P(LD|PS) = 5.66% P(LD|PS) = 0.7940 P(LD|PS) = 12.5% P(LD|PS) = 0.1588 You work for the spy agency CONTROL, and their arch enemy is KAOS, which is incorporated in Delaware for tax reasons. Recruiting agents is hard because not many people are qualified. Only about 1 in 397 people are qualified. There is a screening test. About 95% of the time, a qualified individual will pass the test, and about 3% of the time, an unqualified individual will get lucky and also pass the test. If a random person is chosen to take the test and he or she passes, what is the probability that the person is qualified? Round your answer off to the nearest tenth of a percent. 98.9% 93.7% 7.4% 2.0% Suppose that your friend has a loaded coin that comes up heads 80% of the time and another coin that looks just like the weighted one but is fair. Your friend pulls a coin at random out of his pocket and offers you a bet where he gets $1 if the heads side appears. You are a bit uncomfortable with this and insist on testing the coin. You flip it 5 times, and the heads side comes up once. Given this new information, how likely is it that you have the fair coin? Express your answer as a percentage. Hint: You can use =BINOM.DIST() in Excel to compute the probabilities of the coin flips. 93% 96% 82% 97% A company is considering using a survey company that offers some sample information which may not be very accurate for better decision making for a value of $3,000 in this example. They have provided the following data for all alternatives that are consider here. Question 1 What is the revised probability of having the high demand (HD) for a negative response (NS)? P(HD|NS) = 16% P(HD|NS) = 0.3715 P(HD|NS) = 0.0340 P(HD|NS) = 0.5945 Suppose that your friend has a loaded coin that comes up heads 70% of the time and another coin that looks just like the weighted one but is fair. Your friend pulls a coin at random out of his pocket and offers you a bet where he gets $1 if the heads side appears. How much must you get for a tail for this bet to be fair? $1.50 $1.60 $1.15 $1.00 A company is considering using a survey company that offers some sample information which may not be very accurate for better decision making for a value of $3,000 in this example. They have provided the following data for all alternatives that are consider here. Question 5 What is the probability of having a positive response (PS) for the median demand (MD)? P(PS|MD) = 0.125 P(PS|MD) = 40% P(PS|MD) = 0.84 P(PS|MD) = 60% Suppose that your friend has a loaded coin that comes up heads 80% of the time and another coin that looks just like the weighted one but is fair. Your friend pulls a coin out of his pocket and offers you a bet where he gets $1 if the heads side appears. Your estimate of the probability that he has pulled the loaded coin is 99.999%. You are a bit uncomfortable with this, and insist on testing the coin. You flip it 5 times, and the heads side comes up once. Given this new information, how likely is it that you have the fair coin? Express your answer as a percentage rounded off to the nearest whole number. Hint: You can use =BINOM.DIST() in Excel to compute the probabilities of the coin flips. 0% 78% 94% 84% A company is considering using a survey company that offers some sample information which may not be very accurate for better decision making for a value of $3,000 in this example. They have provided the following data for all alternatives that are consider here. Question 7 What is the revised probability of having the median demand (MD) for a positive response (PS)? P(MD|PS) = 0.1588 P(MD|PS) = 28.30% P(MD|PS) = 0.0473 P(MD|PS) = 60% Suppose that your friend has a loaded coin that comes up heads 70% of the time and another coin that looks just like the weighted one but is fair. Your friend pulls a coin at random out of his pocket and offers you a bet where he gets $1 if the heads side appears. You are a bit uncomfortable with this and insist on testing the coin. You flip it 10 times, and heads come up 4 times. Given this new information, how likely is it that you have the fair coin? Round your answer off to the nearest thousandth. Hint: You can use =BINOM.DIST() in Excel to compute the probabilities of the coin flips. 0.302 0.434 0.666 0.848 Suppose that your friend has 3 coins, 2 loaded and one fair. Coin A comes up heads 70% of the time, coin B is fair, and coin C comes up tails 70% of the time. Your friend pulls a coin at random out of his pocket and offers you a bet where he gets $1 if the heads side appears. You are a bit uncomfortable with this and insist on testing the coin. Without knowing any more, you assess the probability of getting any one of the coins as 1/3. Your friend lets you test the coin by flipping it once, and it comes up heads. Given this new information, what probability do you assign each coin? Round your answer off to the nearest thousandth. A 0.667 B 0.333 C 0.167 A 0.527 [Show More]

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