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This problem offers a meaningful context in which algebraic fractions and tree diagrams can help explain a surprising result in a probability problem. Collaborative working makes it possible for students to tackle an otherwise unmanageable task.
This problem featured in an NRICH video in June 2020.
This problem follows on from the problem Odds and Evens.
"Imagine you had a bag containing a set of balls with whole numbers on them. You choose two balls, and find the total. Can you find a set of balls where the chance of getting an odd total is equal to the chance of getting an even total?"
Click below for a numerical approach suitable for students without experience working algebraically with tree diagrams.
Click below for an algebraic approach using tree diagrams.
You might like to use the Odds and Evens Interactivity to show the experimental probabilities for different sets of up to nine numbers. You can click on the purple cog to change the sets of numbers - just list the numbers you want to use, separated by a space, as in the screenshot
below:
How can you decide if a game is fair?
Take plenty of time to work on the original Odds and Evens problem before trying this extension task.
The problem In a Box offers another context for exploring exactly the same underlying mathematical structure, and could be used as a follow-up problem a few weeks after working on this one.