Why do this
This problem offers an engaging context in which to develop
students' understanding of experimental and theoretical
probability. The interactivity offers students a chance to
get a 'feel' for the situation. Then they can calculate theoretical
probabilities, perhaps by listing at first but then moving towards
multiplying fractions based on conditional probabilities.
Ideally, it would be good to have access to the Lottery
Simulator interactivity as you introduce and work on this problem.
If this is not possible, you could simulate the lottery
yourself by having numbered balls or digit cards in a
To begin with, you could set up the interactivity so that six
balls are available in the 'Number Tumbler' and 'your ticket' has
three numbers. Explain the way this lottery works to the
group and invite suggestions for the numbers to choose on your
ticket. Ask students to predict their chances of winning, and then
simulate some draws - the simulator can be set to draw 10, 20, 50,
100 or even more draws.
"Did we win as often as you expected? How could we calculate
the probability of winning?"
Give students some time to work on this. Some students may
list combinations (systematically or otherwise), whereas others may
use tree diagrams.
Bring the class together to share methods. Highlight anyone
who has listed systematically to discuss the importance of making
sure every combination is considered. If appropriate, move students
towards a tree diagram approach, perhaps referring to the ideas in
Move students on to the follow-up questions to consider the
chances of winning with two or four balls (from six). Ideally, they
will work on this using both listing and tree diagram approaches.
Take time to discuss the symmetry that emerges from choosing a
number on either side of three, and ask students to consider why
Working on this should give them enough confidence using tree
diagrams to be able to answer the remaining follow-up questions
without needing to list combinations.
Did we win as often as you expected?
How could we calculate the probability of winning?
Why is the probability of winning the two from six lottery the same
as the probability of winning the four from six lottery?
Students can be directed here
about the UK National Lottery and see some animations based on real
Students could consider the probability of matching two out of
the three numbers they have chosen, and use the interactivity to
compare experimental with theoretical probabilities. This extends
to considering the UK National Lottery where prizes are awarded for
matching 3, 4 or 5 out of the six numbers drawn.
Some students may wish to extend the 10-ball lottery question to
the general case: what is the hardest lottery to win with $n$
balls? This could be a good opportunity to introduce them to
factorial notation and the binomial coefficients.
It might be helpful for students to have access to the
interactivity in pairs.
To help students to list systematically, start with the two
from six lottery rather than the three from six.