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Coin Tossing Games

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?

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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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Thank Your Lucky Stars

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?

Last One Standing

Age 14 to 16 Challenge Level:

Why do this problem?

This problem is one of a set of problems about probability and uncertainty. Intuition can often let us down when working on probability; these problems have been designed to provoke discussions that challenge commonly-held misconceptions. Read more in this article.
Unusual events are expected to happen to someone if the population is large enough. It's impossible to predict in advance who it will happen to, but after the event we shouldn't be surprised that it has occured. This problem provides an example of this phenomenon.

Possible approach

Before the lesson, prepare a sealed envelope with the word FIVE written on it (assuming you have a class of about 30).
Ask the class to stand up and to each flip a coin. Ask people who flipped tails to sit down. Ask people who flipped heads to flip again. Repeat until there's only one person standing.
Ask the last person standing how many heads they flipped, and reveal with a flourish your prediction - there's a good chance that it will be right, or very close!
"How did I know the last person standing would flip about five heads in a row?"
"Why didn't I try to predict who it would be?"
Give the class a short while to think about these questions and then discuss them with their partner, and then the whole class.
"Imagine we repeated this exercise in the school hall with around 250 students (or 1000). How many heads do you think the last one standing would have flipped?"
Give the class a short while to think about this question before asking them to justify their predictions.
Then use the animation (set up for 256 or 1024) to test out their suggestions - this is a good opportunity to alert learners to the fact that a single trial will not always reflect the theoretical probability, and to discuss the importance of repeating an experiment and taking an average when working with experimental probability. 

There are lots of discussions that can come from this task and from watching the animation at each stage. For example:
  • Why does the number of people standing halve at each stage?
  • Is it possible to predict where the last person standing will be on the grid?
The animation could be used with each learner choosing a point or region of the grid and seeing if they chose correctly, to capture the idea that the unexpected will happen to someone almost every time, but that it's difficult to predict who it will happen to.
There are some suggested questions at the end of the problem that could be used to explore the ideas further. Alternatively, the class could be asked to think of other examples where very unlikely events happen in very large populations.

Key questions

What proportion of the people standing do we expect to sit down on each flip of the coin?
Can we predict how often we should expect an event to occur?
Can we predict to whom we expect the event to happen?

Possible extension

Same Number! offers another opportunity to work with probabilities in order to explain unexpected events.
Students might be interested in this article and related materials on the Understanding Uncertainty website.

Possible support

Encourage learners to start by analysing what happens with only a very small number of people in the room, and to use the interactivity to model it.