Or search by topic
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.
You and I play a game involving successive throws of a fair coin. Suppose I pick HH and you pick TH. The coin is thrown repeatedly until we see either two heads in a row (I win) or a tail followed by a head (you win). What is the probability that you win?
A gambler bets half the money in his pocket on the toss of a coin, winning an equal amount for a head and losing his money if the result is a tail. After 2n plays he has won exactly n times. Has he more money than he started with?
A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand corner of the grid?