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What Does Random Look Like?

Stage: 3 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 working on probability; these problems have been designed to provoke discussions that challenge commonly-held misconceptions. Read more in this article.
This problem requires learners to make sense of experimental data and graphical representations.
The probabilities associated with coin flipping allow learners to analyse and explain the distributions that emerge, and get a feel for the features they would expect a random sequence to exhibit.

Possible approach

Hand out two of these strips to each learner. Ask everyone to make up a sequence of Hs and Ts as if they came from a sequence of coin flips, and to write it down on their first strip, writing "made up" lightly in pencil on the back of the strip. Then ask everyone to flip a coin twenty times and record each outcome on the second strip, writing "real" on the back.
Arrange the learners in groups of three or four, and ask each group to swap ALL their strips with another group, and then challenge them to sort the strips into two piles, "real" and "made up", WITHOUT looking at the back of the strips.
Once every group has had a chance to do this, they can turn over the strips to see how many they got right. Take some time to discuss any criteria they used to decide. 
If you have access to computers:
Ask learners to work in pairs at the computer and give them time to explore the animation. Then ask them to generate several sequences of twenty coin flips and try to get a feel for the features they would expect a random sequence to have. If necessary, suggest that they consider averaging the number of runs of length 2, 3, 4 and so on.
If you don't have access to computers:
If possible, show the interactivity to the whole class and ask them to try to make sense of the bar chart. If this isn't possible, write up a random sequence generated with a coin, and demonstrate the way that the interactivity counts runs.
Ask each group to plot graphs in the same way for the randomly generated sequences they created at the start of the lesson. Once they have done this, encourage them to compare their graphs and identify the key features, perhaps suggesting that they find the average number of runs of length 2, 3, 4 and so on within their group.

Bring the class together and discuss the key features of the random sequences that they found, as well as any explanations of why the run lengths were distributed the way they were, referring to the probabilities of $\frac{1}{2}$ and $\frac{1}{4}$ and so on associated with coin flipping.

Finally, ask each group to give their original real and made-up strips to a DIFFERENT group from the one they swapped with before. Can they use their new-found insights to spot the fakes successfully?

Key questions

What proportion of the time would you expect to flip the same as you got on the previous flip?
What proportion of the time would you expect to flip the same as you got on the TWO previous flips?

Possible extension

The problem Can't Find a Coin challenges learners to fool the computer with a sequence of 100 coin flips.

Possible support

Give learners lots of time to explore the interactivity and make sense of the different bars on the bar chart.