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Win or Lose?

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?

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Fixing the Odds

You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. How should you distribute the balls between the two bags so as to make the probability of choosing a red ball as small as possible and what will the probability be in that case?

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Scratch Cards

To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize?

Do You Feel Lucky?

Stage: 3 and 4 Challenge Level: Challenge Level:1

Why do this problem?

This problem is one of a set of problems about probability and uncertainty. Intuition can often let us down when we meet probability in real life contexts; this problem has been designed to provoke discussions that challenge commonly held misconceptions such as the Gambler's Fallacy.

Possible approach

Hand out this worksheet (Word, PDF) with the statements from the problem, and give everyone time to read them through and decide for themselves whether they think the advice is good or not.
 
Then arrange the class into pairs, and ask half the pairs to come up with arguments in favour and half to come up with arguments against following the advice. Once each pair has had time to rehearse their arguments, arrange them in groups of four with one pair arguing in favour and one pair arguing against, until they reach a consensus.
 
Next, bring the class together for a discussion about how they would respond to each piece of advice and the mathematics they used to justify their decisions.
 
Finally, ask everyone to read through the statements on their own again, to see if they have changed their original views about any of the advice. Those who have changed their minds could explain to the rest of the class which arguments they found particularly persuasive.
 
In the statements, it has been deliberately left ambiguous whether coins are fair or not and so on, so it may be that there is no "right" answer that can be agreed on. The important point is for learners to discuss intelligently the probability in the situations and challenge some of the popular misconceptions that arise.  

Key questions 

Are future results affected by previous results?
Lots of advice is based on accurate statistical data - does that necessarily mean it is useful advice?

Possible extension

Ask learners to collect over several weeks some examples of probability misconceptions in the media, in school or at home, which could be used to create a classroom display.
 
The stage 5 problem Discussing Risk and Reward provides more prompts for discussion about probability in the real world.

Possible support

The problem has been structured as a discussion task so that learners can support each other in coming to a better understanding. By allocating a view for each pair to argue, it allows those who hold these misconceptions the chance to freely explore them without fear of ridicule.