1. CCSS Content Standards G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). S.MD.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). Mathematical Practices 1 Make sense of problems and persevere in solving them. 4 Model with mathematics.

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

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

4. Example 3 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.

5. Example 3 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. 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. 0.35

6. Concept

7. 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. 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.

8. Example 4 Calculate Expected Value

9. Example 4 Calculate Expected Value Answer:

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

11. Example 4 Calculate Expected Value Answer:

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

13. Example 4 A.5.4 B.6.3 C.7.9 D.8.7 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.

14. Example 4 A.5.4 B.6.3 C.7.9 D.8.7 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.

15. Example 4 A.5.9 B.6.6 C.7.1 D.8.3 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.

16. Example 4 A.5.9 B.6.6 C.7.1 D.8.3 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.