[Math Lair] Rockets Candy

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[Rockets]Rockets are a type of candy, presumably made mostly of sugar, that you may see around Hallowe'en. I bet you didn't think they were mathematical though. Well, they are.

Here are some questions involving clustering and probability that relate to the candy. The first set of questions are fairly elementary. The second set require some knowledge of probability and so are a bit more involved.

To answer the questions, you'll need to know that each roll of Rockets contains 15 pieces of candy, and the pieces of candy come in 6 colours: white, pink, orange, yellow, green, and blue. Also, while I'm not sure exactly how Rockets are packaged, assume that each colour is equally likely to occur, that the order is random, and that the colour of each piece of candy is independent of any others in the roll.

Question Set 1

  1. What is the average number of pieces of candy of one single colour that will be found in one roll of Rockets?
  2. Without looking at any rolls of Rockets, write down a few possible distributions of colours in a roll that seem reasonable to you. You might want to record your results in a table. For example, for one possible distribution, you might write:
    White0
    Pink1
    Orange2
    Yellow3
    Green4
    Blue5
  3. Take a six-sided die, and roll it 15 times. Let each roll represent the colour of a piece of candy in a roll, and let white=1, pink=2, ... blue=6. Record the number of each colour that comes up. Repeat a few times and record your results similar to how you did in the previous question.
  4. Now the fun part: Open up a few rolls of Rockets. For each roll, record the distribution of colours similar to how you did above (eat the candy afterwards).
  5. Compare the distributions you created in questions 2 and 3 with the distributions you found in question 4. Do the distributions in question 2 or the ones in chapter 3 appear to be more similar to those in question 4?

Afterwards, you may want to read the page on the clustering illusion, particularly the section about Belief in the Law of Small Numbers.

Question Set 2

  1. What is the probability that there will be no white candies in a roll of Rockets?
  2. What is the probability that there will be no candies of at least one colour in a roll of Rockets? (Hint: The answer is a little more involved than taking the solution to the previous problem and multiplying it by 6)
  3. Write a computer program to simulate a large number of rolls of Rockets, and determine what percentage of them are missing at least one colour. Does the result agree with the answer to the previous question?
  4. Take several packages of Rockets. Determine what percentage of them are missing at least one colour. Does this agree with the results you obtained in the previous questions? If not, why might this not be the case?

Answers to question set 2 can be found on the Rockets answer page.