The eighth graders at Takadona Middle School are organizing a Fun Night for all students in grades 7 and 8. There will be games, movies, and food for the students to enjoy. Each student who comes to Fun Night will receive one red coupon and one green coupon. Some of the red coupons will have a star, which can be turned in for a free hot dog. Similarly, some of the green coupons will have a star, which is good for a free drink. If a coupon does not have a star, it is good for a discount on a food or drink purchase.
It is possible to do a simulation to estimate the chance that a student gets both a free hot dog and a free drink. Instead of actually making coupons and handing them out, the class decides to use two different colored number cubes: a red one and a green one.
Each roll of the number cubes generates a pair of numbers. The outcome of the red number cube is for the red coupons; the outcome of the green one is for the green coupons.
3. Describe how the outcomes of the number cubes can represent the stars for a free meal and a free drink.
You are going to generate 100 pairs of numbers with the two number cubes to simulate 100 students arriving at Fun Night.
4. Design a chart that will make it easy to record the results. The chart should show clearly what each student gets: a free hot dog, a free drink, both, or none.
Use the two different colored number cubes and the chart you designed in problem 4.
Try a few rolls with the number cubes to make sure that your chart works.
Generate 100 pairs of numbers with the number cubes and record the results in your chart. Every possible pair of outcomes on the number cubes should fall into one of the possibilities on your chart.
5. Use your simulation results to estimate:
a. the chance that a student gets a free hot dog;
b. the chance that a student gets a free drink; and
c. the chance that a student gets a free hot dog and a free drink.
6. How close was your answer to problem 2 to the results of the simulation?
8. a. Copy and complete the chance tree.
b. How many students receive a “free meal” consisting of both a free hot dog and a free drink?
c. From your chance tree, what is the chance of receiving a free meal?
d. Reflect How does the chance that you found in part c compare to the chance you found in problem 2 and the one that you found using the simulation in problem 5?
e. Reflect If you started the chance tree about Fun Night with 300 students instead of 120, would your answers for c change? Why or why not?
Combining Situations
E
120 Students
20 Free Hot Dog
100 No Free Hot Dog 5
6 1
6
120 Students
20 Free Hot Dog
100 No Free Hot Dog
Free Drink No Free Drink Free Drink No Free Drink 1
2
1 2
1 2 5 6 1
6
1 2
Dianne complains that the simulation takes too long. She suggests using a chance tree like the one shown to repre-sent the problem.
Suppose 120 students attend Fun Night. Then 20 students will get free hot dogs, and 100 will not get free hot dogs.
7. Explain how you would find the numbers 20 and 100.
Each student who gets a free hot dog has a chance of 1
2 to get a free drink as well.
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10
12
i. ii. iii.
1 6
5 6
1 6
5 6
1 2
1 2
9. a. How many students does each diagram represent? Why do you think this number is chosen?
b. What do grids ii and iii show in terms of the tickets for Fun Night?
c. Copy grid iii into your notebook and shade the portion of the diagram that represents the students who get a free meal.
d. How do the diagrams help you find the fractional part of the students who get a free meal?
e. How does this help you find the chance of a free meal (hotdog and drink)?
Kiesha wanted to use another model to find the chance. She thought an area model might work. This is the series of drawings Kiesha made for the situation on Fun Night.
The organizers of Fun Night are worried about cost. They have decided to change the number of tickets with stars so that the chance of getting a free hot dog will be1
8 and the chance of a free drink will be1
3. 10. a. Draw an area model to represent this situation. Think about a
good number of small squares to use!
b. Use your area model to find the chance of getting both a free hot dog and free drink in this new situation.
Combining Situations
E
One of the events for Fun Night is a student volleyball tournament.
The Fun Night Committee plans to give the players on the winning team shirts donated by the Takadona Sports Apparel Mart.
There will be both tank tops and T-shirts.
Two-thirds (2
3) of all the shirts will be T-shirts.
Logos will be applied randomly to 14of all shirts.
After the tournament, the winning captain can pull one shirt out of the box.
11. What is the chance that the shirt will be a T-shirt with a logo? Show the method you used to find your answer.
One way to find the chance of an event is to list all the possible results and count them, but this is often very time-consuming. Here is a rule that seems to work: The chance for a combination of two events to occur is the chance of the first event times the chance of the second event. We can call this a multiplication rule for chance.
12. a. Does the rule work for problems 10 and 11?
b. Reflect How does this rule show up in the chance tree and the area model?
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Suppose the Takadona Sports Apparel Mart has discovered that the logos do not fit very well on the tank tops. They realize that they can only put logos on T-shirts. Two thirds of the shirts will still be T-shirts, and 1–4of all of the shirts will still have logos
13. a. Choose a total number of shirts. Make a diagram to show how many of these are tank tops, how many are T-shirts, and how many are T-shirts with logos.
b. If you select one of these shirts at random, what is the chance that it will be a T-shirt with a logo?
c. Does this problem follow the multiplication rule? What makes this situation different from the situation in problem 11?
Overall, if there is no connection between getting one outcome and getting the other, then you can use the multiplication rule to find the chance that both events will happen. Otherwise, you have to use some other method to find the chance, like you did in problem 13.