Chapter 19 - Simulation Page 1 Chapter 19 Simulation We learned some basic rules of probability in our discussions of Chapters 17 and 18. We computed the probability of simple events, as well as the probability of unions and intersections of simple events. We computed these probabilities based on a probability model. How did we get these probability models? 1. Empirical Probability – proportion of times the event will occur in the long-run based on proportion of times the event occurs in a long series of repetitions 2. Theoretical Probability – proportion of times the event will occur in the long run based on a set of theories holding true Oftentimes, in practice, it is not practical (or possible) to compute the probability of complicated events – either through theoretical or empirical methods. In these cases, we use a technique called simulation. Definition Using random digits from a table or from computer software to imitate chance behavior is called SIMULATION. Simulation • Is widely used by scientists and engineers to find probabilities in complex situations • Is only as good as the probability model you start with • Uses the fact that the proportion of repetitions on which an event occurs will eventually be close to the probability, so we get good estimates Example 1 Find the probability of getting a run of at least 3 consecutive “heads” or at least 3 consecutive “tails” in the experiment where a fair coin is tossed 10 times. Step 1 Give a probability model. Each toss has a probability of 0.5 of a “heads” and 0.5 of a “tails” Tosses are independent of one another (recall: independence tells us that the outcome of one toss of the coin does not change the probability of the outcome of any other toss) Step 2 Assign digits to represent outcomes Goal: assign digits in a way that matches the probabilities from Step 1 Solution: (one of several possibilities) • Let one digit represent 1 toss of the coin • an odd digit represents heads and an even digit represents tails this works because each digit in the table has a 1/10 chance to be the next digit and successive digits are independent, so 5/10 = half the digits are odd and 5/10 = half the digits are evenChapter 19 - Simulation Page 2 Step 3 Simulate many repetitions We’ll do three together. Now, you do one repetition. CLICKER Use your clicker to record whether you had a run of at least three “heads” or at least three “tails” in your one repetition. Did you have a run of at least 3 “heads” or at least 3 “tails” A. YES B. NO So, our estimate of the probability of getting a run of at least 3 “heads” or at least 3 “tails” our of 10 coin flips isChapter 19 - Simulation Page 3 How do we assign random digits to carry out simulation? Example 2 a. Choose a person at random from a group of which 70% are employed. b. Choose one person at random from a group where 73% are employed. c. Choose one person at random from a group where 50% are employed, 20% are unemployed, and 30% are not in the labor force.Chapter 19 - Simulation Page 4 Example (not in text) When would we want to simulate a coin flip rather than actually flip a coin? When we discussed response error, we discussed how when respondents are asked “sensitive” questions, they may not be truthful in their response. One such question that has most likely been asked of you in the past (in Junior High or High School) concerns marijuana use. Many of you admitted to not being truthful on these types of questions. One method that attempts to get more accurate responses uses randomization. In some studies, respondents are assured a level of anonymity by allowing a coin flip to dictate a “forced response”. In this type of an experiment, respondents are asked to flip a coin. Only the respondent knows the outcome of the coin flip. If the coin turns up “heads”, respondents answer to the sensitive question is automatically “yes”. If the coin comes up “tails” respondents are asked to answer the yes or no question truthfully. This was, the respondents are assured that there is no way their response of “yes” can be traced back to whether it was a forced “yes” or a true “yes”. Some researchers believe this experiment works best when the coin flipping is simulated…why? Let’s use this idea to answer the following question in class now: Do you currently smoke marijuana on a regular basis at least once a month? Use the random digits table for your simulation of a coin flip. Choose a “random” place to enter the random digits table and make an odd number be “heads” and an even number “tails”. Then, let “heads” mean you automatically answer “yes” and let “tails” mean you tell the truth when answering the question. CLICKER A. “YES” (either because you flipped “heads” or you flipped “tails” and are answering truthfully) B. “NO” (only answer “no” if you really don’t use marijuana on a regular basis) Now, what is a good way to estimate the proportion of those who use marijuana at least once a month? What are the drawbacks to this method? What are the benefits?Chapter 19 - Simulation Page 5 More Elaborate Simulations Example 4 A couple plans to have children until they have a girl or until they have three children. What is the probability they will have a girl among their children? Simulation Step 1 Probability Model Step 2 Assign Digits Step 3 Repeat many Simulations True Probability Tree Diagram A useful tool for organizing more complicated probabilities in a graphical form is called a tree diagram. A tree diagram represents different stages of a probability model by different “branchings”. Multiplication up the branches denotes intersections of events (considering conditions) and adding the resulting intersections’ probabilities can give us the probability of a union. Now compute the true probability this couple will have a girl.Chapter 19 - Simulation Page 6 Example 5 Suppose a patient has been told he needs a kidney transplant. He would like to know the probability of surviving for at least five more years. He is told that 90% survive the transplant operation. Of those that survive the operation, 60% of transplants are successful and the other 40% need to go back on dialysis. The proportion that survive for at least five