How to make clear tasks for students about resolving paradoxes? Question

When creating questions for coursework, exams, and similar, I sometimes want to task students with explaining counter-intuitive results or dissolving paradoxes, as I consider this is a good opportunity for them to train or showcase their understanding of the bigger picture or their skills to make an argument.

Now, a problem with just asking “Why is our intuition wrong here?” or “Solve this paradox.” is that it requires that everybody agrees upon what is intuitive or paradoxical here. A student may (quite rightfully) consider the correct result intuitive or just be happy with calculating the correct result and noticing that it is different from what intuition assumes.

Therefore I loathed such questions as a student since they strongly depend on interpretation. Can I somehow avoid this problem without dropping the respective task altogether?

Example: The Birthday Paradox

The birthday paradox is that n = 23 is the smallest number such that for a group of n people, it is more likely that two birthdays coincide than not. The number n is much lower than what most people would expect intuitively. I am seeking to ask a question that has an answer along the lines of:


  While the probability that a given member’s birthday coincides with another only grows slowly (linearly) when the group size is increased, birthdays can also coincide between newly added group members. The number of such pairings grows faster (quadratically).


Now if I ask the students: “Why is n so low?”, they could ask: “In relation to what?”. Also, they might ‘simply’ calculate n, and I could not rightfully complain that this is not what the question asked for. (Of course, correctly calculating n in this case requires including pairings between newly added nodes and hopefully grants the students the kind of insight I am aiming at. However, some may still apply some formulas blindly and this is not so obvious in more complicated examples.)
Let your students debunk a wrong argument

Present a specific faulty argument based on intuition. Then let your students find the flaw in it. This way you relieve your students from the burden to divine what you consider intuitive. They also have a more specific task that cannot be solved by just deriving the correct result.

For example a task for the birthday paradox could be:


  Bob argues:
  
  
    A random person has a different birthday from me with a probability of 364/365 (ignoring leap years, seasonal variations, etc.). Hence, if I am in a group with n−1 other people, the probability that all of them have a different birthday is (364/365)n−1. This probability is first smaller than 0.5 for n = 254.
  
  
  Find the flaw in Bob’s argument.