Why do this problem?
This problem gives practice in a relatively complicated
probability calculation drawn from a simple situation. It will
require clear visualisation of the possibilities, accurate working
and a good understanding of permutations and combinations. It gives
a good chance to practise calculator skills and the use of the
$^nC^r$ button. The context will allow a discussion concerning the
role of intuition versus calculation in probability.
Before beginning this problem ask students whether they think
that a weaker player is more or less likely to win over a long
match. Can they explain why they believe that this is the case? Are
these arguments completely convincing to the rest of the class. Can
they estimate how likely they believe that the weaker player is to
win over $11$ frames? Perhaps each student could make a guess. The
closest guess without going above wins.
In probability, calculation is used to settle disputes. Before
students solve the problem they will need to be very clear as to
the combinations of matches which lead to a win for the weaker
player. They will then need to write this down in a clear way
before working out the numbers. Spreadsheets or calculators will be
necessary for this. As the calculation is relatively long it will
require good calculator or recording skills successfully to obtain
the answer, even if the route to the answer is conceptually clear
to students. It might provide an opportunity explicitly to practise
the use of calculator keys such as ANS, $^nC^r$ and $!$. Can
students encode the entire calculation in a single line of
Once the answer is found, compare with the initial estimates
that the class made. Whose intuition was reliable?
Before attempting the computation, can you estimate the chance
of the weaker player winning the match?
If the weaker player only had a 4% chance of winning, what do
you feel would happen to his chances of winning the match as the
number of frames increases?
Which player must win the final frame of the match?
What are the possible final scorelines?
Can students write down a generalisation of the formula used
to show the chance of a player winning a match played over $2n+1$
frames? The numerical investigation will allow students to put
their formula into practice.
Reduce the complexity of the calculation: what is the chance
of winning a best of $3$ or a best of $5$ match?