# Sweet Probability (Investigation)

Lesson

## Objectives

• To practice with finding theoretical and experimental  probability.
• To practice with replacement and non-replacement probability.
• To practice with independent and dependent events.

## Materials

• Varied amounts of 4 different types of candies (for example, have differing amounts of Hershey’s Bars, Twix Bars, Milky Way Bars, and Snickers Bars)
• A bag that you cannot see through

## Procedure

Work on your own or in small groups.

### PART 1

Before placing the candy in the bag answer the following questions:

1. How much of each kind of candy do you have?
2. For each type of candy determine the theoretical probability it will be selected from the bag.
3. Which candy are you most likely to pick from the bag? Why?
4. Which candy has a probability closest to zero of being chosen? Why? Does this mean it is likely to be chosen?
5. List the candies in order of least to most likeliness.
6. Are there any two candies that are equally likely to be chosen? Why or why not?

### PART 2

1. Now place the candy into the bag. Shake up the bag.
2. Reach into the bag and pull out one piece of candy. Record what it was and place it back into the bag.
3. Shake up the bag again.
4. Reach into the bag again and pull out one piece of candy. Record what it was and place it back into the bag. Do this until you have recorded a total of 20 candies.
##### QUESTIONS

1. Make a frequency chart for the data you have just collected. It should look similar to the following table:

Candy Frequency
Hershey's Bar 10
Twix 5
Milky Way Bars 2
Snickers Bars 3

2. From the frequency chart you have just created answer the following:

1. What is the experimental probability of selecting each candy?
2. Pick your favorite two candies. What is the experimental probability of getting either of these candies?
3. How many of each candy would you expect to get if you had originally started out with 100 total candies in the bag? What about for $500$500 total candies? Why? Explain how you arrived at your answer.
4. Does the first candy you drew from the bag effect what was drawn second from the bag? In other words, was the drawing of the candy an independent or dependent event? Why or Why not?
5. Was this experiment done with or without replacement? Is this connected with the dependence or independence of the events? Why or why not?
6. Compare the experimental probability of drawing each candy from the bag with the theoretical probability of drawing each candy from the bag (calculated in Part 1). What do you notice? Why do you think that is?
7. Compare with a friend! Did they get the same experimental probability as you for each candy? Why or why not? Explain why you think this is.
8. Work with a friend! Combine your data with as many people as possible and create a new frequency table. Re calculate the experimental probability for each candy. Compare this to the theoretical probabilities calculated in Part 1. What do you notice?

### Part 3

1. Suppose now that you have the situation where instead of putting the candy back in the bag after each time you put the candy to the side. You do not actually need to do it this time.
2. Assign each of  your candy brands a number 1-4. For example I could label my candy brands (1)Hershey’s Bars, (2)Twix Bars, (3)Milky Way Bars, (4)Snickers Bars.
##### Questions:
1. Suppose you draw two candies from the bag at random and do not replace them. What is the probability:
2. You get two candies in a row that correspond to the brand labeled (1)
3. You choose a candy from (1) followed by a candy from (3)?
4. A candy from (2) is not selected?
5. How many possible different candy combinations can be chosen from the bag if you select three candies?
6. Is the action of selecting a candy at random dependent on the candy chosen previously in this scenario? Why or why not?
7. How many possible different candy combinations can be chosen from the bag if you select 5 candies? Put all the candy into the bag and randomly pull out 5 different candies. After selecting each from the bag  place the candy to the side. Which combination did you end up with?
8. Compare with a friend! What combination did they end up with when they drew out 5 candies? Was it the same as yours?
9. What do you notice about independent and dependent events? Are they related to replacement and non-replacement? If so, how? Use the work you did in Part 2 and Part 3 to justify your answer.
10. Make some observations about independent and dependent events.
11. Work with a friend! Brainstorm at least 3 other situations that would be considered dependent events.