Why do this problem?
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.
This problem combines an element of experimentation with some analysis to explain unexpected results. Although the probabilities involved can't be calculated, estimated probabilities can be derived through experiment.
Arrange learners in groups of 4, with a pack of cards and some counters for each group. Ask them to remove the Jacks, Queens and Kings, and then shuffle the remaining cards. Explain how to set up the 'snake' and then ask them to put a different counter on each of the first four cards. Once they have chosen a counter each, get them to see how far they can each go before falling off the
Bring the class together and ask a few groups what happened on their table. Express surprise at how many counters finished on the same card. Collect each group's result on the board in a tally chart like the one below:
|All counters on the same card
|Three counters on the same card
|Two pairs of counters on two cards
|Two counters on the same card
|All counters on different cards
"Maybe this only happened because we only did the experiment a few times... I'd like each group to repeat the activity at least 10 times and keep a record of your results"
Once the class have accumulated sufficient results, gather the results in the tally chart. Discuss any reasons the learners can suggest as to why the counters so often ended up on the same card.
Use the class's results to estimate the probability of each outcome. Set each group the challenge of developing a way of using the activity as a fundraising game, using the experimental probabilities to decide the pricing structure and winning conditions for their game. These could be presented to the rest of the class at the end of the lesson, pitching to be the group chosen to represent
the class at the next School Fair, and explaining why their game would raise the most funds.
Finally, allow some time for learners to work on the challenge of finding a snake where all four (and then five) counters end up on different cards.
Where did your counters end up?
Why do the counters so often end up in the same place?
Is it possible to find 'snakes' of cards where all four counters end up in different places?
The initial task should be accessible to all. When planning a way to use the activity as a fundraiser, less confident learners could be encouraged to restrict the winning options. For example, they could explore pricing structures for a game that only pays out when all participants land on different cards.
Retiring to Paradise provides a different context for considering the importance of spread as well as average when working with data.