LESSON 13–4 Simulations

Five-Minute Check (over Lesson 13–3) Then/Now New Vocabulary Key Concept: Designing a Simulation Example 1: Design a Simulation by Using a Geometric Model Example 2: Design a Simulation by Using Random Numbers Example 3: Conduct and Summarize Data from a Simulation Key Concept: Calculating Expected Value Example 4: Calculate Expected Value Lesson Menu

A. B. C. D. 5-Minute Check 1

A. B. C. D. 5-Minute Check 2

A. B. C. D. 5-Minute Check 3

Camla knows the bus she needs comes every hour and a half Camla knows the bus she needs comes every hour and a half. What is the probability that Camla waits 15 minutes or less for the bus? A. B. C. D. 5-Minute Check 4

Find the probability that a point chosen at random from inside the circle lies in the shaded region. 5-Minute Check 5

A spinner has 5 equal sections that are red, blue, red, blue, red A spinner has 5 equal sections that are red, blue, red, blue, red. What is the probability of the pointer landing on blue? A. 20% B. 40% C. 60% D. 80% 5-Minute Check 6

You found probabilities by using geometric measures. Design simulations to estimate probabilities. Summarize data from simulations. Then/Now

probability model simulation random variable expected value Law of Large Numbers Vocabulary

Concept

Step 2 Our simulation will consist of 40 trials. Design a Simulation by Using a Geometric Model BASEBALL Maria got a hit 40% of the time she was at bat last season. Design a simulation that can be used to estimate the probability that she will get a hit at her next at bat this season. Step 1 Possible Outcomes Theoretical Probability Maria gets a hit 40% Maria gets out (100 – 40)% or 60% Step 2 Our simulation will consist of 40 trials. Example 1

Design a Simulation by Using a Geometric Model Step 3 One device that could be used is a spinner divided into two sectors, one containing 40% of the spinner’s area and the other 60%. To create such a spinner, find the measure of the central angle of each sector. Get a hit: 40% of 360° = 144° Get out: 60% of 360° = 216° Step 4 A trial, one spin of the spinner, will represent one at-bat. A successful trial will be a hit and a failed trial will be getting out. The simulation will consist of 40 trials. Example 1

GAME SHOWS You are on a game show in which you pull one key out of a bag that contains four keys. If the key starts the car in front of you, then you have won it. Which of the following geometric spinners accurately reflects your chances of winning? A. B. C. D. Example 1

Design a Simulation by Using Random Numbers PIZZA A survey of Longmeadow High School students found that 30% preferred cheese pizza, 30% preferred pepperoni, 20% preferred peppers and onions, and 20% preferred sausage. Design a simulation that can be used to estimate the probability that a Longmeadow High School student prefers each of these choices. Step 1 Possible Outcomes Theoretical Probability Cheese 30% Pepperoni 30% Peppers and onions 20% Sausage 20% Example 2

Design a Simulation by Using Random Numbers Step 2 We assume a student’s preferred pizza type will fall into one of these four categories. Step 3 Use the random number generator on your calculator. Assign the ten integers 0–9 to accurately represent the probability data. The actual numbers chosen to represent the outcomes do not matter. Outcome Represented by Cheese 0, 1, 2 Pepperoni 3, 4, 5 Peppers and onions 6, 7 Sausage 8, 9 Example 2

Design a Simulation by Using Random Numbers Step 4 A trial will consist of selecting a student at random and recording his or her pizza preference. The simulation will consist of 20 trials. Example 2

PETS A survey of Mountain Ridge High School students found that 20% wanted fish as pets, 40% wanted a dog, 30% wanted a cat, and 10% wanted a turtle. Which assignment of the ten integers 0–9 accurately reflects this data for a random number simulation? A. Fish: 0, 1, 2 B. Fish: 0, 1 Dog: 3, 4, 5, 6 Dog: 2, 3, 4, 5 Cat: 7, 8 Cat: 6, 7 Turtle: 9 Turtle: 8, 9 C. Fish: 0, 1 D. Fish: 0, 1 Dog: 2, 3, 4, 5, 6 Dog: 2, 3, 4, 5 Cat: 7, 8 Cat: 6, 7, 8 Turtle: 9 Turtle: 9 Example 2

Maria got a hit 40% of the time she was at bat last season. Conduct and Summarize Data from a Simulation BASEBALL Refer to the simulation in Example 1. Conduct the simulation and report the results, using the appropriate numerical and graphical summaries. Maria got a hit 40% of the time she was at bat last season. Make a frequency table and record the results after spinning the spinner 40 times. Example 3

This is an experimental probability. Conduct and Summarize Data from a Simulation Based on the simulation data, calculate the probability that Maria will get a hit at her next at-bat. 0.35 This is an experimental probability. The probability that Maria makes her next hit is 0.35 or 35%. Notice that this is close to the theoretical probability, 40%. So, the experimental probability of her getting out at the next at-bat is 1 – 0.35 or 65%. Make a bar graph of these results. Example 3

Conduct and Summarize Data from a Simulation Example 3

Which one of these statements is not true about conducting a simulation to find probability? A. The experimental probability and the theoretical probability do not have to be equal probabilities. B. The more trials executed, generally the closer the experimental probability will be to the theoretical probability. C. Previous trials have an effect on the possible outcomes of future trials. D. Theoretical probability can be calculated without carrying out experimental trials. Example 3

Concept

First, calculate the geometric probability of landing in each region. Calculate Expected Value ARCHERY Suppose that an arrow is shot at a target. The radius of the center circle is 3 inches, and each successive circle has a radius 5 inches greater than that of the previous circle. The point value for each region is shown. A. Let the random variable Y represent the point value assigned to a region on the target. Calculate the expected value E(Y) for each shot of the arrow. First, calculate the geometric probability of landing in each region. Example 4

Calculate Expected Value Example 4

Answer: The expected value of each throw is about 4.21. Calculate Expected Value Answer: The expected value of each throw is about 4.21. Example 4

Calculate Expected Value ARCHERY Suppose that an arrow is shot at a target. The radius of the center circle is 3 inches, and each successive circle has a radius 5 inches greater than that of the previous circle. The point value for each region is shown. B. Design a simulation to estimate the average value or the average of the results of your simulation of shooting this game. How does this value compare with the expected value you found in part a? Example 4

Calculate Expected Value Assign the integers 0–324 to accurately represent the probability data. Region 10 = integers 1–9 Region 8 = integers 10–64 Region 5 = integers 65–169 Region 2 = integers 170–324 Use a graphing calculator to generate 50 trials of random integers from 1 to 324. Record the results in a frequency table. Then calculate the average value of the outcomes. Example 4

Calculate Expected Value Answer: The average value 5.62 is greater than the expected value 4.21. Example 4

A. In a similar situation to Example 4a, if the following are the geometric probabilities of a target with 3 regions, what is the expected value? Assume each region is worth the value it is named. A. 5.4 B. 6.3 C. 7.9 D. 8.7 Example 4

B. If the chart is populated by data from a simulation and each region is worth the value it is named, calculate the average value from these 50 trials. A. 5.9 B. 6.6 C. 7.1 D. 8.3 Example 4

LESSON 13–4 Simulations