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.

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

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

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.

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

Concept

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.

Example 4 Calculate Expected Value

Example 4 Calculate Expected Value Answer:

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

Example 4 Calculate Expected Value Answer:

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

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.

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.

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.

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.