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Why do this problem?

This problem offers an engaging context in which to develop students' understanding of theoretical probability. They can calculate theoretical probabilities, perhaps by listing at first, but then by moving towards multiplying fractions based on conditional probabilities.

Possible approach

You could introduce the problem by simulating a lottery using numbered balls or digit cards in a bag.  

Explain the way this lottery works to the group and invite them to choose three numbers. Ask students to predict their chances of winning, and then try a few draws to see if anyone wins.
 
"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 this article.
 
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 this happens.
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.
 

Key questions

How often would you expect to win?

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?

Possible support

 
To help students to list systematically, start with the two from six lottery rather than the three from six.
 

You may want to experiment with this lottery simulator before moving on to the theoretical probabilities

 

 
 

Possible extension

Students can be directed here to read about the UK National Lottery.
 
Students could consider the other probabilities in 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.