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Name: ______________________________ Date: March 16, 2017 7th Grade Math: Unit: Statistics and Probability Packet 103: Simulations Homeroom: Objectives: Scholars will be able to learn how to perform simulations to estimate probabilities. Scholars will be able to use various devices to perform simulations (e.g., coin, number cubes, cards). Scholars will be able to compare estimated probabilities from simulations to theoretical probabilities. Agenda: 1. 2. 3. 4. 5. Review Do Now Simulations Setting up Simulations Guided Practice Running a Simulation Do Now: 6. Basketball Player 7. Independent Practice 8. Exit Ticket 9. Homework Overview Homework 103: Simulations 1. You roll a single die numbered from 1 to 6 twice. What is the probability of rolling a 6 the first time and an odd number the second? 2. A jar contains 12 caramels, 7 mints and 16 dark chocolates. What is the probability of selecting a dark chocolate and then a caramel without replacing the first piece of candy? 1 Simulations In previous lessons, you estimated probabilities of events by collecting data empirically or by establishing a theoretical probability model. There are real problems for which those methods may be difficult or not practical to use. A simulation is a procedure that will allow you to answer questions about real problems about real problems by running experiments that closely resemble the real situation. It is often important to know the probabilities of real-life events that may not have known theoretical probabilities. Scientists, engineers, and mathematicians design simulations to answer questions that involve topics such as disease, water flow, climate change, or functions of an engine. Results from the simulations are used to estimate probabilities that help researchers understand problems and provide possible solutions to these problems. Setting up Simulations How likely is it that a family with three children has all boys or all girls? Let’s assume that a child is equally likely to be a boy or girl. Instead of observing the results of actual births, a toss of a fair coin could be used to simulate a birth. If the toss results in heads (H), then we could say a boy was born; if the toss results in tails (T), then we could say a girl was born. 1. How could a number cube be used to simulate getting a boy or girl birth? 2. How could a deck of cards be used to simulate getting a boy or girl birth? 2 Guided Practice There are five steps in completing a simulation. Let’s go through the five steps of a simulation by using the children example! 1. Define the outcome of the REAL experiment: 2. Chose a simulation device: a. Suppose that a family has three children. How could you use a coin to simulate the gender of a child? b. How could you use a desk of card to simulate the gender of three children? 3 3. Define what is meant by a trial. Link it to the real experiment: 4. Define what is meant by a success in the trial. Link it to the real experiment: 5. Perform the trials! More is better in this case! a. Suppose 100 trials are performed, and in those 100 trials, 28 resulted in either HHH or TTT. What is the estimated probability that a family of 3 children has either three boys or three girls? b. What is the estimated probability that a family of three would have some combination other than three boys or three girls? 4 Running a Simulation 1. Find an estimate of the probability that a family with three children will have exactly one girl using the following outcomes of 50 trials of tossing a fair coin three times per trial. Using H to represent a boy birth and T to represent a girl birth. HHT HHT THT THH HTH HTH TTT HHH HHT HTH HHH HHT THH TTT THT TTH TTH HTT TTH TTH THT HHH HTH HTT TTT THT HTH TTT THH HHT HTT THH HTT HTT HHT HHH TTT HHH HTH THT TTH THT TTH TTT TTT HHH THT THT HHH HTT 2. Perform a simulation of 50 trials by rolling a fair number cube in order to find an estimate of the probability that a family with three children will have exactly one girl. a. Specify what outcome of one roll of a fair number cube will represent a boy, and what outcome will represent a girl. b. Simulate 50 trials, keeping in mind that one trial requires three rolls of the number cube. List the results of your 50 trials. 5 c. Calculate the estimated probability. 3. Calculate the theoretical probability that a family with three children will have exactly one girl. a. Use a tree diagram to list the possible outcomes for a family with three children. First Child Second Child Third Child Outcome b. Assume having a boy and having a girl are equally likely. Calculate the theoretical probability that a family with three children will have all boys or all girls. c. Compare it to the estimated probabilities found in parts 1 and 2 above. 6 Basketball Player Suppose that, on average, a basketball player makes about three out of every four foul shots. In other words, she has a 75% chance of making each foul shot she takes. Since a coin toss produces equally likely outcomes, it could not be used in a simulation for this problem. Instead, a number cube could be used by specifying that the numbers 1, 2 or 3 could represent a hit, and number 4 represent a miss, and the number 5 and 6 would be ignored. Based on the following 50 trials of rolling a fair number cube, find an estimate of the probability that she makes 5 or 6 of the foul shots she takes. 441323 124144 121411 344221 131224 143143 214411 343214 121142 331113 342124 333434 321341 222442 213344 243224 423221 123131 321442 311313 442123 243122 111422 343123 321241 323443 311423 242124 121423 211411 422313 232323 114232 122111 311214 324243 142141 141132 443431 433434 441243 224341 414411 322131 241131 214322 411312 343122 214433 323314 7 Independent Practice A mouse is placed at the start of the maze shown below. If it reaches station B, it is given a reward. At each point where the mouse has to decide which direction to go, assume that it is equally likely to go in either direction. At each decision point, 1, 2 or 3, it must decide whether to go left (L) or right (R). It cannot go backwards. 1. Create a theoretical model of probabilities for the mouse to arrive at terminal A, B and C. a. List the sample space for the paths the mouse can take. For example, if the mouse goes left at decision point 1 and then right at decision point 2, then the path would be denoted LR. b. Are the paths in your sample space equally likely? c. What are the theoretical probabilities that a mouse reaches terminal points A, B, and C? Explain. 8 2. Based on the following set of simulated paths, estimated the probabilities that the mouse arrives at point A, B, C. RR RL LL RR RR LR LR LR RL LR LR RR LL RR LL RL LR RL RR LR RL LR RR LL LR RR RL RL LL LL LL LR LR RL RR RL RR RL LR LL a. How do you reach point A? What is the experimental probability of making it there? b. How do you reach point B? What is the experimental probability of making it there? c. How do you reach point C? What is the experimental probability of making it there? d. How do these estimated probabilities compare to the theoretical probabilities above? 9 10 Name: ______________________________ Date: March 16, 2017 7th Grade Math: Unit: Statistics and Probability Packet 103: Simulations Homeroom: Mrs. Petrozzi: (614) 725-9693 Redo: Score: ________/5 _________% DNG A B C D F Directions: Answer the following questions. Show all of your work and circle your final answer. Suppose that a dartboard is made up of the 8 X 8 grid of squares shown below. Also, suppose that when a dart is thrown, it is equally likely to land on any one of the 64 squares. A point is won if the dart lands on one of the 16 black squares. Zero points are earned if the dart lands in a white space. 1. For one throw of the dart, what is the probability of winning a point? Note that a point is won if the dart lands on a black square. 2. For one throw of the dart, what is the probability of NOT winning a point? Note that you do not earn any points if the dart lands on a white square. 11 Suppose a game consists of throwing a dart three times. A trial consists of three rolls of the number cube. (1 is a win, 2, 3, or 4 is a loss). The results of the experiment are listed below. 324 321 144 113 421 332 332 322 223 222 411 112 421 333 222 322 433 414 414 112 124 412 111 212 113 224 443 242 431 212 221 322 244 233 413 241 424 222 314 341 111 412 331 212 442 223 433 224 241 324 3. For each roll, you can either win 0, 1, 2 or 3 points. For each outcome listed in the table, how many points did you win? 324 332 411 322 124 224 221 241 111 223 321 332 112 433 412 443 322 424 412 433 144 322 421 414 111 242 244 222 331 224 113 223 333 414 212 431 233 314 212 241 421 222 222 112 113 212 413 341 442 324 4. You rolled the number cube 50 times. Write the probabilities for getting 0, 1, 2, or 3 points. Number of Points Times You Rolled Probability 0 1 2 3 12 Name: ______________________________ Date: March 16, 2017 Homeroom: 7th Grade Math: Unit: Statistics and Probability Packet 103: Simulations Directions: Answer the following questions. Circle your final answers. When you finish the exit ticket, place it face down on your desk and assign yourself to your advanced work or an independent reading book. Nathan is your school’s soccer player. When he takes a shot on goal, he scores half of the time on average. To estimate probabilities of the number of goals Nathan makes, use a simulation with a number cube. One roll of a number cube represents one shot. 1. Decide what outcome of a number cube you want to represent a goal scored by Nathan in one shot. 2. For this problem, what represents a trial of taking 6 shots? 3. What is the theoretical probability of Nathan getting a goal when he takes a shot? 13
