Binomial Conditions

Why do this problem?

It is common for students who have studied the binomial distribution to be quite unfamiliar with the conditions necessary for this to be an appropriate distribution to model an experiment.  The constant probability one is fairly straightforward, but the independence condition is quite poorly understood.  Even some textbooks fail to describe the conditions correctly.

In this problem, students are asked to construct scenarios in which one of the two conditions holds but the other does not.  The first is easier, but the second requires a clear understanding of the term "independent".  Students are likely to deepen their understanding of this concept by working on this problem, as well as gain a deeper appreciation for the need for these conditions.  It will help students to be able to distinguish between situations which are described by a binomial distribution and those which are not.

Note that this problem does not address the need for the number of trials to be fixed from the start, and for the random variable to be the total number of successes.

Possible approach

This problem will be most helpful after students have had some exposure to the binomial distribution and have developed a familiarity with it in "regular" situations.  The teacher could first ask their students if they can recall the conditions for a binomial distribution to be appropriate, and remind them if they have forgotten.  The teacher should then check that the students understand the terms in the conditions, and in particular the term "independent".  Once this is clear, students could work on their own initially to construct such examples, and then share their ideas with a partner before feeding back to the whole class.

There might be significant confusion around the idea of the probability of a dependent event.  If the second trial is not independent of the first one, then the probability of success on the second trial will change after the first trial has been performed.  So when we say that "each trial has an equal probability of success", what we mean is "before the experiment begins, each trial has equal probability of success".  If instead we meant "regardless of what happens on earlier trials, each trial has an equal probability of success", we would be saying that the trials are independent and they each have an equal probability of success, which is just (i) and (ii) together.  This point may well come up through discussion.

Key questions
 

What does "independent" mean?
How could the separate trials be best represented?
What are some key features of the binomial distribution probabilities?

Possible extension

How many different scenarios can you construct where the binomial distribution fails to be the correct distribution, even though it has some of the features required?

Possible support

If students have not yet worked on Binomial or Not?, this might be a useful starting point. 

For the question of independence, it may be easier to come up with reasons why two trials are dependent.  If you can't do that, then they are likely to be independent.
Limit yourself to just two trials.  How could these be represented?  (At least two different ways!)
What would the probabilities have to be if the number of successes is described by the binomial distribution?  How can you break this?