INTR STATISTCL REASONING
INTR STATISTCL REASONING STAT 110
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Date Created: 10/26/15
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 longrun based on proportion oftimes 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 oftheories 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 ls widely used by scientists and engineers to nd probabilities in complex situations ls 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 05 ofa heads and 05 ofa 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 0 Let one digit represent 1 toss ofthe coin c an odd digit represents heads and an even digit represents tails this works because each digit in the table has a 110 chance to be the next digit and successive digits are independent so 510 half the digits are odd and 510 half the digits are even Chapter 19 Simulation Page 1 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 N0 80 our estimate of the probability of getting a run of at least 3 heads or at least 3 tails our of 10 coin ips is Chapter 19 Simulation Page 2 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 0 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 3 Example for fun not in text When would we want to simulate a coin ip rather than actually ip 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 ofyou admitted to not being truthful on these types of questions Forced Yes Method Respondents ip a coin Only the respondentknows the outcome ofthe coin flip lfthe coin turns up heads respondents answer yes to the sensitive question lfthe coin comes up tails respondents are asked to answer truthfully This way 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 ipping is simulatedwhy Let s use this idea to answer the following question in class now Do you currently smoke marijuana on a regular basis de ned as at least once a month Use the random digits table for your simulation of a coin ip heads means FORCED YES tails means TRUTHFUL YES or NO CLICKER A YES either because you flipped heads or you flipped tails and are answering truthfully B NO you flipped tails and you really don t use marijuana on a regular basis Now what is a good way to estimate the proportion ofthose who use marijuana at least once a month What are the drawbacks to this method What are the bene ts Chapter 19 Simulation Page 4 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 ofa 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 ofa union Now compute the true probability this couple will have a girl Chapter 19 Simulation Page 5 Example 5 Suppose a patient has been told he needs a kidney transplant He would like to knowthe probability of surviving for at least ve more years He is told that 90 survive the transplant operation Ofthose 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 ve years is 70 for those with a new kidney that works and 50 for those that have to return to dialysis Step 1 Probability Model Step 2 Assign digits to outcomes Step 3 Simulate Many repetitions True Probablity Chapter 19 Simulation Page 6