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High School Statistics and Probability
Domain Conditional Probability and the Rules of Probability
Cluster Understand independence and conditional probability and use them to interpret data
Standard Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. <span class='clarification'>For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.</span>
Task But mango is my favorite…

But mango is my favorite…


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Alignments to Content Standards: S-CP.A.5 S-CP.B S-MD.B.7
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Task

A seven-year-old boy has a favorite treat, Super Fruity Fruit Snax.

These "Fruit Snax" come in pouches of 10 snack pieces per pouch, and the pouches are generally sold by the box, with each box containing 4 pouches.

The snack pieces come in 5 different fruit flavors, and usually each pouch contains at least one piece from each of the 5 flavors. The website of the company that manufactures the product says that equal numbers of each of the 5 fruit flavors are produced and that pouches are filled in such a way that each piece added to a pouch is equally likely to be any one of the five flavors.

Of all the 5 fruit flavors, the seven-year-old boy likes mango the best. One day, he was very disappointed when he opened a pouch and there were no (zero) mango flavored pieces in the pouch. His mother (a statistician) assured him that this was no big deal and just happens by chance sometimes.

  1. If the information on the company's website is correct,

    1. What proportion of the population of snack pieces is mango flavored?
    2. On average, how many mango flavored pieces should the boy expect in a pouch of 10 snack pieces?
    3. What is the chance that a pouch of 10 would have no mango flavored pieces? Was the mother's statement reasonable? Explain. (Hint: if none of the 10 independently selected pieces are mango, then all 10 pieces are "not mango.")
    4. The family then finds out that there were in fact no mango flavored pieces in any of the 4 pouches in the box they purchased. Again, if the information on the company's website is correct,
      1. What is the chance that an entire box of 4 pouches would have no mango flavored pieces? (Hint: How is this related to your answer to question (iii) in part (a)?)
      2. Based on your answer and based on the fact that this event of an entire box with "no mangoes" happened to this family, would you be concerned about the company's claims, or would you say that such an event is not surprising given the company's claims? Explain.

IM Commentary

This task builds on students' previous work with probability in Grade 7—specifically with standards 7.SP.7 and 7.SP.8.

This task can be used as an instructional task to develop students’ understanding of independence and students’ ability to calculate the probability of intersection events using the multiplication rule for independent events (S.CP.5 and S.CP.ClusterB). Students also use a computed probability to evaluate a claim (S.MD.7). This task can also be used to illustrate standard 3 of the mathematical practices.

As you discuss this task with your students, be sure they understand that the bags are filled in a way that results in each of the ten pieces put into a pouch being equally likely to be any of the five flavors. Begin by having students think about the questions in part (a). Encourage students to think about the event described in question (iii) as the intersection of ten events—first piece not mango, second piece not mango, and so on.

In part (b), make sure that students see how the probability calculated in part (a) relates to what they are being asked to do in question (i) of part (b).

As students answer question (ii) in part (b), they are also gaining experience with mathematical practice standard 3 (construct viable arguments and critique the reasoning of others).

Solution

    1. If each of the 5 flavors occurs equally in the population, the proportion of mango flavored pieces is $\frac15$ or .20.
    2. If the proportion of mango flavored pieces in the extremely large population is $\frac15$ or .20, then a random sample of 10 pieces (a pouch) should have $.20 \cdot 10 = 2$ mango flavored pieces on average.
    3. The chance of selecting a single non-mango flavored piece is $\frac45$ or .80. If the selection of each piece is independent, then the chance of getting 10 non-mango flavored pieces is or about .1074. So yes, the mother's statement that a single pouch might not have any mango flavored pieces seems reasonable, happening in slightly more than 1 out of 10 pouches.
      1. An entire box would consist of 40 snack pieces. As above, the chance of selecting a single non-mango flavored piece is $\frac45$ or .80. If the selection of each piece is independent, then the chance of getting 40 non-mango flavored pieces is or about .0001. Also, a student could use the answer determined in 1c above for the probability of a single "mango-less" pouch (.1074) and raise that value to the fourth power to represent 4 pouches, $.1074\cdot.1074\cdot.1074\cdot.1074 = \text{ about } .0001$.
      2. Most students would say that such an event is too rare to occur by chance (1 in 10,000 boxes) if the company's claims are true. Therefore, the company's claims may need to be called into question. However, some students who are familiar with sweepstakes and offers from children's cereals, snacks, etc. may say that a 1 in 10,000 chance is not that unusual and is more likely than winning a grand prize, etc. The important thing is that the student's decision should be explained carefully and with reference to the answer to Question b.i.

But mango is my favorite…

A seven-year-old boy has a favorite treat, Super Fruity Fruit Snax.

These "Fruit Snax" come in pouches of 10 snack pieces per pouch, and the pouches are generally sold by the box, with each box containing 4 pouches.

The snack pieces come in 5 different fruit flavors, and usually each pouch contains at least one piece from each of the 5 flavors. The website of the company that manufactures the product says that equal numbers of each of the 5 fruit flavors are produced and that pouches are filled in such a way that each piece added to a pouch is equally likely to be any one of the five flavors.

Of all the 5 fruit flavors, the seven-year-old boy likes mango the best. One day, he was very disappointed when he opened a pouch and there were no (zero) mango flavored pieces in the pouch. His mother (a statistician) assured him that this was no big deal and just happens by chance sometimes.

  1. If the information on the company's website is correct,

    1. What proportion of the population of snack pieces is mango flavored?
    2. On average, how many mango flavored pieces should the boy expect in a pouch of 10 snack pieces?
    3. What is the chance that a pouch of 10 would have no mango flavored pieces? Was the mother's statement reasonable? Explain. (Hint: if none of the 10 independently selected pieces are mango, then all 10 pieces are "not mango.")
    4. The family then finds out that there were in fact no mango flavored pieces in any of the 4 pouches in the box they purchased. Again, if the information on the company's website is correct,
      1. What is the chance that an entire box of 4 pouches would have no mango flavored pieces? (Hint: How is this related to your answer to question (iii) in part (a)?)
      2. Based on your answer and based on the fact that this event of an entire box with "no mangoes" happened to this family, would you be concerned about the company's claims, or would you say that such an event is not surprising given the company's claims? Explain.
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