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Why do this problem?
This problem offers an engaging context in which to discuss probability and uncertainty. Intuition can often let us down when working on probability; this problem has been designed to provoke discussions that challenge commonly-held misconceptions. You can read more about it in this article.
This problem requires students to make sense of experimental data. The probabilities associated with coin flipping allow students 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 1
Hand out two of these strips to each student. 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 students 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.
Using the interactive coin flipper, or a spreadsheet generating random numbers, or actual coins, invite students to generate sequences of coin flips and draw their attention to the length of runs in randomly generated sequences. How often does a coin flip give the same result as it did in the previous coin flip? How many runs of length three might you expect? How many of four? and
so on.
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?
Possible approach 2
Hand out two of these strips to each student. 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 students in groups of three or four, and ask each group to swap ALL their strips with another group, and tell them you will be challenging them to sort the strips into two piles, "real" and "made up", WITHOUT looking at the back of the strips.
Using the interactive coin flipper, or a spreadsheet generating random numbers, or actual coins, invite students to generate sequences of coin flips and analyse the length of runs in randomly generated sequences. How often does a coin flip give the same result as it did in the previous coin flip? How many runs of length three might you expect? How many of four? and so on.
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.
Then challenge them to use their insights 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.
Finally, ask each group to create another set of real and made-up strips and ask each group to swap ALL their strips with the other group, and as before, challenge them to sort the strips into two piles, "real" and "made up". Can the students use their new-found insights to 'disguise' 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 support
Encourage students to record what the longest run length is for each sequence of 20 in the interactivity.
Possible extension
The problem Can't Find a Coin invites students to analyse sets of 100 coin flips to see whether a sequence is truly random.