This problem is one of a set of problems about probability and
uncertainty. Intuition can often let us down when working on
probability; these problems have been designed to provoke
discussions that challenge commonly held misconceptions. Read more
in this
article.

An important strategy in answering probability questions
requires us to consider whether it is easier to work out the
probability of an event occurring or NOT occurring, and using the
fact that the two probabilities sum to 1. This problem uses this
technique.

For a class of 30, ask everyone to write down a number between
$1$ and $225$ without letting
anyone see. (If the class size is $N$, ask them to write
down a number between $1$ and $(\frac{N}{2})^2$)

Once everyone has chosen, go round the class and ask each person to read out their number, with everyone listening to hear if their number is read out again. On most occasions, there will be duplicates.

"OK, let's have another go, you have $225$ numbers to choose from so let's see if you can manage to make them all different!" (Don't allow collaboration on which numbers to choose!)

The experiment can be repeated a few times, or the interactivity can be used to generate sets of random numbers for a class of 30 to get a sense of how often everyone manages to pick a unique number.

On one occasion, after the first person has read their number, interrupt and ask "What's the probability that the next person has written a different number?", and highlight that it's very likely! Ask a similar question a little further along the line: "What's the probability that the next person has a number that has not been read out so far?" Though the fraction will have changed, it will still be very close to 1, so learners' intuition may be that it is very likely that everyone will choose a different number.

Once everyone has had the chance to be surprised that duplicate numbers happen much more often than not, hand out this worksheet (Word, PDF) and ask the class to discuss the questions and work out their answers with their partner. (You may wish to detach the Extension task from the bottom of the sheet, and hand it out later to those who need it.)

The Extension task is the well-known Birthday Problem; learners could work out the probability of shared birthdays in various class sizes and then use school records to see if the tutor groups in the school have shared birthdays with the expected frequency.

What's the probability (at each stage) that the next person
has written a number that has not been read out so far?

What's the probability that the second AND third person (and
fourth, and fifth...) have numbers different from the first person
and each other?

If I know the probability that every number is different, how
can I work out the probability that at least 2 numbers are the
same?

The Birthday Problem provides an interesting extension.

Students might be interested in these articles and related
materials on the Understanding Uncertainty website:

For interested students (or teachers), the "Matching Problems"
section of this
article by David Spiegelhalter explains the derivation of
the $(\frac{N}{2})^2$ formula mentioned above - the maths
required to fully understand it is at Stage 5 level.