What Does Random Look Like?
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.
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.
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.
Key questions
What proportion of the time would you expect to flip the same as you got on the previous flip?
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.
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