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Using Simulation to Estimate a Probability

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Examples, videos, and solutions to help Grade 7 students learn how to design their own simulations using colored disks and a random number table.

New York State Common Core Math Grade 7, Module 5, Lesson 11

Download worksheets for Grade 7, Module 5, Lesson 11

Lesson 11 Student Outcomes

• Students design their own simulations.
• Students learn to use two more devices in simulations: colored disks and a random number table.

Lesson 11 Summary

In the previous lesson, you carried out simulations to estimate a probability. In this lesson, you had to provide parts of a simulation design. You also learned how random numbers could be used to carry out a simulation.
To design a simulation:
• Identify the possible outcomes and decide how to simulate them, using coins, number cubes, cards, spinners, colored disks, or random numbers.
• Specify what a trial for the simulation will look like and what a success and a failure would mean.
• Make sure you carry out enough trials to ensure that the estimated probability gets closer to the actual probability as you do more trials. There is no need for a specific number of trials at this time; however, you want to make sure to carry out enough trials so that the relative frequencies level off.

Lesson 11 Classwork

Example 1: Simulation
In the last lesson, we used coins, number cubes, and cards to carry out simulations. Another option is putting identical pieces of paper or colored disks into a container, mixing them thoroughly, and then choosing one.
For example, if a basketball player typically makes five out of eight foul shots, then a colored disk could be used to simulate a foul shot. A green disk could represent a made shot, and a red disk could represent a miss. You could put five green and three red disks in a container, mix them, and then choose one to represent a foul shot. If the color of the disk is green, then the shot is made. If the color of the disk is red, then the shot is missed. This procedure simulates one foul shot.

Exercises 1–2
1. Using colored disks, describe how one at-bat could be simulated for a baseball player who has a batting average of 0.300. Note that a batting average of 0.300 means the player gets a hit (on average) three times out of every ten times at bat. Be sure to state clearly what a color represents.
2. Using colored disks, describe how one at-bat could be simulated for a player who has a batting average of 0.273. Note that a batting average of 0.273 means that on average, the player gets 273 hits out of at-bats.

Example 2: Using Random Number Tables
Why is using colored disks not practical for the situation described in Exercise 2? Another way to carry out a simulation is to use a random-number table, or a random-number generator. In a random-number table, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 occur equally often in the long run. Pages and pages of random numbers can be found online.
For example, here are three lines of random numbers. The space after every five digits is only for ease of reading. Ignore the spaces when using the table.
To use the random-number table to simulate an at-bat for the 0.273 hitter in Exercise 2, you could use a three-digit number to represent one at bat. The three-digit numbers from 000-272 could represent a hit, and the three-digit numbers from 273-999 could represent a non-hit. Using the random numbers above, and starting at the beginning of the first line, the first three-digit random number is 252, which is between 000 and 272, so that simulated at-bat is a hit. The next three-digit random number is 566, which is a non-hit.

Exercise 3
3. Continuing on the first line of the random numbers above, what would the hit/non-hit outcomes be for the next six at-bats? Be sure to state the random number and whether it simulates a hit or non-hit.
Example 3: Baseball Player
A batter typically gets to bat four times in a ballgame. Consider the 0.273 hitter of the previous example. Use the following steps (and the random numbers shown above) to estimate that player’s probability of getting at least three hits (three or four) in four times at-bat.
a. Describe what one trial is for this problem.
b. Describe when a trial is called a success and when it is called a failure.
c. Simulate trials. (Continue to work as a class, or let students work with a partner.)
d. Use the results of the simulation to estimate the probability that a 0.271 hitter gets three or four hits in four times at-bat. Compare your estimate with other groups.

Example 4: Birth Month
In a group of more than 12 people, is it likely that at least two people, maybe more, will have the same birth-month? Why? Try it in your class.
Now suppose that the same question is asked for a group of only seven people. Are you likely to find some groups of seven people in which there is a match, but other groups in which all seven people have different birth-months? In the following exercise, you will estimate the probability that at least two people in a group of seven, were born in the same month.

Exercises 4–7
4. What might be a good way to generate outcomes for the birth-month problem—using coins, number cubes, cards, spinners, colored disks, or random numbers?
5. How would you simulate one trial of seven birth-months?
6. How is a success determined for your simulation?
7. How is the simulated estimate determined for the probability that a least two in a group of seven people were born in the same month?

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