Simulating dice Assume that you have access to a computer

Problem 1BSC Simulating Dice When two dice are rolled, the total is between 2 and 12 inclusive. A student simulates the rolling of two dice by randomly generating numbers between 2 and 12. Does this simulation behave in a way that is similar to actual dice? Why or why not?

Problem 3BSC Simulating Birthdays A student wants to conduct the simulation described in Example, but no calculator or computer is available, so the student uses 365 individual index cards to write the individual numbers between 1 and 365. The student then shuffles the cards, selects one, and records the result. That card is replaced, the cards are again shuffled and a second number is drawn. This process is repeated until 25 birthdays are generated. Does this simulation behave the same way as the process of selecting 25 people and recording their birthdays? Why or why not?

Problem 2BSC Simulating dice Assume that you have access to a computer that can randomly generate whole numbers between any two values. Describe how this computer can be used to simulate the rolling of a pair of dice.

Problem 4BSC Simulating coin Flips One student conducted the simulation described in Example and stated that the probability of getting a sequence of six 0s or six 1s is 0.977. What is wrong with that statement?

Problem 5BSC Describe the simulation procedure. (For example, to simulate 10 births, use a random number generator to generate 10 integers between 0 and 1 inclusive, and consider 0 to be a male and 1 to be a female.) Brand Recognition The probability of randomly selecting an adult who recognizes the brand name of McDonald’s is 0.95 (based on data from Franchise Advantage). Describe a procedure for using software or a TI-83/84 Plus calculator to simulate the random selection of 50 adult consumers. Each individual outcome should be an indication of one of two results: (1) The consumer recognizes the brand name of McDonald’s; (2) the consumer does not recognize the brand name of McDonald’s.

Problem 6BSC: Describe the simulation procedure. (For example, to simulate 10 births, use a random number generator to generate 10 integers between 0 and 1 inclusive, and consider 0 to be a male and 1 to be a female.) Lefties - Ten percent of people are left-handed. In a study of dexterity, 15 people are randomly selected. Describe a procedure for using software or a TI-83/84 Plus calculator to simulate the random selection of 15 people. Each of the 15 outcomes should be an indication of one of two results: (1) Subject is left-handed; (2) subject is not left-handed.

Problem 7BSC Describe the simulation procedure. (For example, to simulate 10 births, use a random number generator to generate 10 integers between 0 and 1 inclusive, and consider 0 to be a male and 1 to be a female.) Shaquille O'Neal Shaquille O’Neal was a professional basketball star who had a reputation for being a poor free-throw shooter. As of this writing, he made 5155 of the 9762 free throws that he attempted, for a success ratio of 0.528. Describe a procedure for using software or a TI-83/84 Plus calculator to simulate his next free throw. The outcome should be an indication of one of two results: (1) The free throw is made; (2) the free throw is missed.

Problem 8BSC Describe the simulation procedure. (For example, to simulate 10 births, use a random number generator to generate 10 integers between 0 and 1 inclusive, and consider 0 to be a male and 1 to be a female.) Simulating Hybridization When Mendel conducted his famous hybridization experiments, he used peas with green pods and yellow pods. One experiment involved crossing peas in such a way that 75% of the offspring peas were expected to have green pods, and 25% of the offspring peas were expected to have yellow pods. Describe a procedure for using software or a TI-83/84 Plus calculator to simulate 20 peas in such a hybridization experiment. Each of the 20 individual outcomes should be an indication of one of two results: (1) The pod is green; (2) the pod is yellow.

Problem 9BSC Develop a simulation using a TI-83/84 Plus calculator, STATDISK, Minitab, Excel, or any other suitable calculator or computer software program. Simulating Brand Recognition Study Refer to Exercise, which required a description of a simulation. a. Conduct the simulation and record the number of consumers who recognize the brand name of McDonald’s. If possible, obtain a printed copy of the results. Is the proportion of those who recognize McDonald’s reasonably close to the value of 0.95? b. Repeat the simulation until it has been conducted a total of 10 times. In each of the 10 trials, record the proportion of those who recognize McDonald’s. Based on the results, do the proportions appear to be very consistent or do they vary widely? Based on the results, would it be unlikely to randomly select 50 consumers and find that about half of them recognize McDonald’s? Exercise Describe the simulation procedure. (For example, to simulate 10 births, use a random number generator to generate 10 integers between 0 and 1 inclusive, and consider 0 to be a male and 1 to be a female.) Brand Recognition The probability of randomly selecting an adult who recognizes the brand name of McDonald’s is 0.95 (based on data from Franchise Advantage). Describe a procedure for using software or a TI-83/84 Plus calculator to simulate the random selection of 50 adult consumers. Each individual outcome should be an indication of one of two results: (1) The consumer recognizes the brand name of McDonald’s; (2) the consumer does not recognize the brand name of McDonald’s

Problem 10BSC Develop a simulation using a TI-83/84 Plus calculator, STATDISK, Minitab, Excel, or any other suitable calculator or computer software program. Simulating Left-Handedness Refer to Exercise, which required a description of a simulation. a. Conduct the simulation and record the number of left-handed people. Is the percentage of left-handed people from the simulation reasonably close to the value of 10%? b. Repeat the simulation until it has been conducted a total of 10 times. Record the numbers of left-handed people in each case. Based on the results, would it be unlikely to randomly select 15 people and find that none of them is left-handed? Exercise Describe the simulation procedure. (For example, to simulate 10 births, use a random number generator to generate 10 integers between 0 and 1 inclusive, and consider 0 to be a male and 1 to be a female.) Lefties Ten percent of people are left-handed. In a study of dexterity, 15 people are randomly selected. Describe a procedure for using software or a TI-83/84 Plus calculator to simulate the random selection of 15 people. Each of the 15 outcomes should be an indication of one of two results: (1) Subject is left-handed; (2) subject is not left-handed.

Problem 11BSC Develop a simulation using a TI-83/84 Plus calculator, STATDISK, Minitab, Excel, or any other suitable calculator or computer software program. Simulating the Shaq Refer to Exercise, which required a description of a simulated free throw by basketball player Shaquille O’Neal. a. Repeat the simulation five times and record the number of times that the free throw was made. Is the percentage of successful free throws from the simulation reasonably close to the value of 0.528? b. Repeat part (a) until it has been conducted a total of 10 times. Record the proportion of successful free throws in each case. Based on the results, would it be unlikely for Shaquille O’Neal to make all of five free throws in a game? Exercise Describe the simulation procedure. (For example, to simulate 10 births, use a random number generator to generate 10 integers between 0 and 1 inclusive, and consider 0 to be a male and 1 to be a female.) Shaquille O'Neal Shaquille O’Neal was a professional basketball star who had a reputation for being a poor free-throw shooter. As of this writing, he made 5155 of the 9762 free throws that he attempted, for a success ratio of 0.528. Describe a procedure for using software or a TI-83/84 Plus calculator to simulate his next free throw. The outcome should be an indication of one of two results: (1) The free throw is made; (2) the free throw is missed.

Problem 12BSC Develop a simulation using a TI-83/84 Plus calculator, STATDISK, Minitab, Excel, or any other suitable calculator or computer software program. Simulating Hybridization Refer to Exercise, which required a description of a hybridization simulation. a. Conduct the simulation and record the number of yellow peas. If possible, obtain a printed copy of the results. Is the percentage of yellow peas from the simulation reasonably close to the value of 25%? b. Repeat the simulation until it has been conducted a total of 10 times. Record the numbers of peas with yellow pods in each case. Based on the results, do the numbers of peas with yellow pods appear to be very consistent? Based on the results, would it be unlikely to randomly select 20 such offspring peas and find that none of them has yellow pods? Exercise Describe the simulation procedure. (For example, to simulate 10 births, use a random number generator to generate 10 integers between 0 and 1 inclusive, and consider 0 to be a male and 1 to be a female.) Simulating Hybridization When Mendel conducted his famous hybridization experiments, he used peas with green pods and yellow pods. One experiment involved crossing peas in such a way that 75% of the offspring peas were expected to have green pods, and 25% of the offspring peas were expected to have yellow pods. Describe a procedure for using software or a TI-83/84 Plus calculator to simulate 20 peas in such a hybridization experiment. Each of the 20 individual outcomes should be an indication of one of two results: (1) The pod is green; (2) the pod is yellow.

Problem 13BSC Probability of a Run of Three Use a simulation approach to find the probability that when five consecutive babies are born, there is a run of at least three babies of the same sex. Describe the simulation procedure used, and determine whether such runs are unlikely.

Problem 14BSC Probability of a Run of Four Use a simulation approach to find the probability that when six consecutive babies are born, there is a run of at least four babies of the same sex. Describe the simulation procedure used, and determine whether such runs are unlikely.

Problem 15BSC Gender-Selection Method As of this writing, the latest results available from the Microsort YSORT method of gender selection consist of 127 boys in 152 births. That is, among 152 sets of parents using the YSORT method for increasing the likelihood of a boy, 127 actually had boys and the other 25 had girls. Assuming that the YSORT method has no effect and that boys and girls are equally likely, simulate 152 births. Is it unlikely to get 127 boys in 152 births? What does the result suggest about the YSORT method?

Problem 16BSC Nasonex Treatment Analysis Nasonex is a nasal spray used to treat allergies. In clinical trials, 1671 subjects were given a placebo, and 2 of them developed upper respiratory tract infections. Another 2103 patients were treated with Nasonex and 6 of them developed upper respiratory tract infections. Assume that Nasonex has no effect on upper respiratory tract infections so that the rate of those infections also applies to Nasonex users. Using the placebo rate of 2/1671, simulate groups of 2103 subjects given the Nasonex treatment, and determine whether a result of 6 upper respiratory tract infections could easily occur. What does that suggest about Nasonex as a cause of upper respiratory tract infections?

Problem 18BB Simulating Birthdays a. Develop a simulation for finding the probability that when 50 people are randomly selected, at least 2 of them have the same birth date. Describe the simulation and estimate the probability. b. Develop a simulation for finding the probability that when 50 people are randomly selected, at least 3 of them have the same birth date. Describe the simulation and estimate the probability.

Problem 17BSC Simulating the Monty Hall Problem A problem that once attracted much attention is the Monty Hall problem, based on the old television game show Let’s Make a Deal, hosted by Monty Hall. Suppose you are a contestant who has selected one of three doors after being told that two of them conceal nothing but that a new red Corvette is behind one of the three. Next, the host opens one of the doors you didn’t select and shows that there is nothing behind it. He then offers you the choice of sticking with your first selection or switching to the other unopened door. Should you stick with your first choice or should you switch? Develop a simulation of this game and determine whether you should stick or switch. (According to Chance magazine, business schools at such institutions as Harvard and Stanford use this problem to help students deal with decision making.)

Problem 19BB Genetics: Simulating Population control A classical probability problem involves a king who wanted to increase the proportion of women by decreeing that after a mother gives birth to a son, she is prohibited from having any more children. The king reasons that some families will have just one boy, whereas other families will have a few girls and one boy, so the proportion of girls will be increased. Is his reasoning correct? Will the proportion of girls increase?