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
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
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?
Can you arrange the cards so that if you place six counters on the
first six cards, they end up on six different cards at the end? If
you can't, prove it can't be done!
Investigate packs of cards with Jacks included (counting as
eleven), Queens (counting as twelve), and Kings (counting as
thirteen). Or packs with fewer suits. Or...
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.