Sample Lesson

Sample Lesson: Course 1, Unit 7, Simulation Models

Simulation and the Law of Large Numbers (Launch)

In 1985 Major League Baseball switched from a best-of-five league playoff series to a best-of-seven league playoff series. In a best-of-five series, the team which first wins three games wins the series.

Think About This Situation
The winners of the league playoffs represent the National and American Leagues in the World Series, which is also a best-of-seven games.
a.	How many games do you have to win to be victorious in a best-of-seven series?
b.	Why do you think Major League Baseball went from a five- to a seven-game championship series?

Investigation 1: Simulating Percentages Using Random Digits (Explore)

In this investigation, you will explore playoff series of various lengths. Of course, if the teams are evenly matched, each team has a probability of 1/2 of winning; but teams of all sorts often compete and they are seldom evenly matched.

The Cyclones are playing the Hornets for the softball championship. Based on their history, the Cyclones have a 60% chance of beating the Hornets in any one game.

If the championship series were only one game long, what is the probability that the better team (Cyclones) would win?

If the Cyclones and the Hornets were to play 100 games, about how many games would you expect the Cyclones to win? The Hornets? Explain your reasoning.

Describe how you would design a simulation model for Part (b) using a table of random digits or random numbers from your calculator. How would you split up the numbers so that the Cyclones have a 60% chance of winning and the Hornets a 40% chance of winning?

Why isn't it appropriate to use a coin flip in your model to determine which team wins a game?

Is it possible to have a best-of-two game series? Explain.

Could an even number of games ever be used for a playoff series? Explain your reasoning.

Suppose the Cyclones and the Hornets play a three-game series.

Design a simulation model to determine the probability that the better team (Cyclones) will win a best-of-three game series.

Conduct your simulation 200 times. Share the work among the groups in your class. Record your data in a frequency table like that below.

Number of Games Won by the Cyclones in a Three-Game Series	Frequency
0
1
2
Total	200

Which rows represent a win of the championship series by the Cyclones? By the Hornets?

What is your estimate of the probability that the Cyclones will win a best-of-three series?

Next explore the idea of a five-game series between the same two teams.

Design a simulation model to determine the probability that the better team (Cyclones) will win a best-of-five game series.

Conduct your simulation 200 times by sharing the work among the groups in your class. Record your data in a frequency table where the number of games won by the Cyclones goes from 0 to 3. Why just to 3?

What is your estimate of the probability that the Cyclones will win a best-of-five series?

Now explore the idea of a seven-game series for the same teams.

Design a simulation model to determine the probability that the better team (Cyclones) will win a best-of-seven game series.

Conduct your simulation 200 times. Share the work among the groups in your class. Place your results in a frequency table showing the number of games won by the Cyclones.

What is your estimate of the probability that the Cyclones will win a best-of-seven series?

Complete the table below using the results from Activities 1, 3, 4, and 5.

Type of Series	Probability the Cyclones Win
One game
Best-of-three
Best-of-five
Best-of-seven

What pattern do you observe in the table? What team, the Cyclones or the Hornets, would have the better chance to win a best-of-nine game series? Estimate the probability of their winning.

Improve your estimate in Part (b) by carrying out a simulation.

From a mathematical point of view, why do you now think that Major League Baseball went from a five- to a seven-game championship series?

Make histograms of your frequency tables from Activities 3-5. How are these histograms alike? How are they different?

Tennis players have two chances to get their serve "in". Monica makes about 50% of her first serves. If she has to try a second serve, Monica makes about 80% of those.

Describe how to use a table of random digits to simulate this situation.

Describe how to use the rand function on your calculator to simulate this situation.

Conduct your simulation once, placing the result in a frequency table. Your frequency table should have three rows: makes first serve, misses first serve and makes second serve, and double-faults (misses both serves).

Repeat your simulation 50 times.

Estimate the probability that Monica double-faults.

In the previous situations, the percentages of success - winning a game or getting a serve "in" - were multiples of 10. In those cases, you may have used single digits from your random digits table or from your calculator. Often, as in the next two situations, the percentages are not "nice" numbers.

9.	The 1992 National High School Senior Survey conducted by the University of Michigan found that 29% of high school seniors had smoked a cigarette within the last 30 days.
	a.	Design a simulation that estimates the number of smokers in a randomly selected group of 20 seniors.
	b.	Perform your simulation ten times, placing the results on a number line plot that shows the number of smokers in each group of 20 selected students.

10.

In the almost 50-year history of National Basketball Association playoffs, the home team has won about 67% of the games. Suppose that the Los Angeles Lakers are playing the Phoenix Suns in the NBA playoffs. The two teams are equally good, except for this home team advantage. The playoffs are a best-of-seven series. The first two games will be played in Phoenix, the next three (if needed) in Los Angeles, and the final two (if needed) in Phoenix.

What is the probability that the Suns will win a game if it is at home? What is the probability that the Suns will win a game if it is played in Los Angeles? What is the probability the Suns will win the first game of the series? The second game? The third game? The fourth game? The fifth game? The sixth game? The seventh game?

Design a simulation model for this situation.

Conduct your simulation 200 times by sharing the work among the groups in your class. Place the results in a frequency table like that shown below.

Number of Games in a Playoff Series Won by the Suns	Frequency
0
1
2
3
4
Total	200

What is the probability that the Suns win the playoffs?

To cut travel costs, suppose the NBA schedules four games in Los Angeles followed by three in Phoenix.

Design a simulation model to determine the probability the Suns win the playoffs in this situation.
Conduct your simulation 200 times. Share the work with other groups.
What is your estimate of the probability the Suns win this series?

Compare the probabilities in Parts (d) and (e). What is your conclusion?

Checkpoint
a.	In playoff series, what is the advantage of a longer series over a shorter one?
b.	How can random numbers be used in simulations when the two outcomes are not equally likely?
c.	Sheila has a 55% chance of winning a ping pong game against Bobby. Describe a simulation model for determining the probability of Sheila winning a best-of-nine series of ping-pong games with Bobby. Should Bobby prefer a best-of-three series?
Be prepared to share your group's thinking and simulation model with the class.

In this investigation, you have explored a variation of the Law of Large Numbers. The Law of Large Numbers says, for example, that if you roll a die more and more times, the proportion of fives tends to get closer to 1/6 . The Cyclones have a 60% chance of winning each game. In a longer series, the percentage of games the Cyclones win tends to be closer to 60% than in a shorter series. And so the Cyclones are more likely to win the longer series.

On
Your
Own

Recall that in Major League Baseball, the World Series is a best-of-seven games series. In Modeling Task 3 on page 10, you estimated the probability that the World Series will go seven games if the teams are equally matched. The probability is 0.3125.

If the two teams aren't evenly matched, do you think the World Series is more likely to go seven games or less likely to go seven games than if the teams are evenly matched? Why?

Suppose that the teams are not evenly matched and that the American League team has a 70% chance of winning each game. Describe a simulation model of the World Series in this case.

Conduct your simulation 5 times. Add your results to a copy of the frequency table below so that there is a total of 100 repetitions of the simulation.

Number of Games Needed in the Series	Frequency
4	24
5	30
6	24
7	17
Total

What is your estimate of the probability that the series will go seven games in the case when one team has a 70% chance of winning each game? Is this probability of a seven-game series more or less than when the teams are evenly matched?

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