Question

A cereal company puts prizes in specially marked boxes of cereal. There are 4 possible prizes, randomly distributed among the boxes so that any box is equally likely to contain any of the 4 prizes. (There is only one prize per box.) One of the prizes is a whistle, and another is a colored pencil.

1. Which of the following is not a binomial random variable in the above scenario? Choose one.

(a) X = number of boxes that contain a whistle out of the next 10 boxes purchased.

(b) X = number of boxes that do not contain a whistle out of the next 10 boxes purchased.

(c) X = number of boxes that will need to be purchased to get 3 whistles.

(d) X = number of boxes that contain either a whistle or a colored pencil out of the next 10 boxes purchased.

Answer 1

Option (c) X = number of boxes that will need to be purchased to get 3 whistles is not a binomial random variable because it does not have a fixed number of trials, which is a requirement for a binomial distribution.

Step-by-step explanation:

The question revolves around identifying which option is not a binomial random variable in the given cereal prize scenario. Criteria for a binomial random variable include only two possible outcomes per trial (success or failure), a fixed number of trials, and independence between the trials.

Option (a) X = number of boxes that contain a whistle out of the next 10 boxes purchased meets the binomial criteria, with 'success' being finding a whistle, and 'failure' not finding one. Option (b) X = number of boxes that do not contain a whistle out of the next 10 boxes purchased is also binomial with the roles of success and failure reversed. Option (d) X = number of boxes that contain either a whistle or a colored pencil out of the next 10 boxes purchased is binomial, with success in finding either prize.

However, option (c) X = number of boxes that will need to be purchased to get 3 whistles is not a binomial random variable because it does not have a fixed number of trials; the number of boxes needed to achieve 3 whistle prizes could vary without limit. This type of scenario would be modeled by a geometric or negative binomial distribution instead.