What does random look like?
What Does Random Look Like? printable sheet
On a strip like the one below, ask a friend to make up a sequence of twenty Hs and Ts that could represent a sequence of heads and tails generated by a fair coin, and ask them to write "made up" lightly in pencil on the back of the strip.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
Then ask your friend to flip a fair coin twenty times and record the results on a second strip, and this time ask them to write "real" lightly in pencil on the back.
Take the two strips and try to work out which was real and which was made up - you could create similar strips and challenge your friend in the same way.
Here is an animation which generates twenty random coin flips.
Use the animation to generate several sequences of twenty coin flips, and try to summarise the features you would expect a random sequence to have.
How would you analyse whether a sequence came from a real coin?
Send us your ideas and justify the method you use to decide.
Are you now better at spotting fakes? Ask your friend to create two more strips and see if you can find the truly random one.
Are you now better at creating fakes? Give your friend two more strips and see if they can spot the fake.
On average, how often will my coin flip give me the same result as it did in my previous coin flip?
On average, how many runs of length 3 might I expect?
How many of length 4?...
Thank you to everyone who got in touch to share their thoughts on this problem. We received solutions from many schools which included Heckmondwike School, Little Ilford School and Wilson's School in the UK, the British International School in The Hague, the British Vietnamese International School of Hanoi, the Frederick Irwin Anglican School and St Phillip's College in Australia, St Vincent's School in France, PSBBMS in India and the Harrow International School in Hong Kong. You seemed to have really enjoyed conducting the experiment and sharing your ideas .
Many of you began by focusing on the probability of tossing either a head or a tail. Kim, from the British International School, shared these thoughts:
Every coin has two sides, heads and tails. If this is a fair coin, then it's 50-50. In my opinion to find the real one, find the one closest to 50-50 in ratio.
Raja, from Little Ilford School, also wondered about the balance of heads and tails:
To tell whether a sequence came from a real coin and a human mind is impossible. However, there are signs which could help someone decipher if a sequence one is real or fake. For example, the probability that a sequence (of 20 flips) that contains exactly an H or T is 1 in a 1048576, which is extremely low. That is a telltale sign that a sequence is fake. What I would expect from a real sequence is about 50% heads and 50% tails because there are only 2 states that a fair coin can be in. If I were to give a range, possibly 45% to 55%, however, remember at the end of the day it all relies on chance.
Sean, who attends Wilson's School, reflected on the sequencing of the heads and tails in his two lists:
My instant reaction when going around this was it should roughly be a fifty-fifty split between the two. Yet when seeing the real version, I realised I would be wrong.
So, when I went back to the drawing board, I realised that there was a common pattern. The more realistic ones are not a “back and forth pattern”. What I mean by this is that there is a low chance that if three heads are in a consecutive row, there is a slim chance there will be three tails in a row. I surmise that one mistake people can make when making a fake string of heads and tails that they are familiar with something in their head so they just write it down again with just a small change. To make it really authentic, you should not do that.
Dilara, who attends the International School in The Hague, also suggested that fake lists tended to include more patterns than real ones. Here's an example that she shared with our team:
Strategies for identifying fake lists featured in several of your responses, including this response submitted by Krithigan from Wilson's School:
I believe that a real coin will not have a specific pattern to it as there were 4Hs followed by 2Ts and then 5Hs [in my list] which shows that you cannot predict what is coming next. However, a fake coin will have a similar pattern as it regularly changed for T to H and vice versa. We think that if we include multiple Hs or Ts it will seem unrealistic but it does the opposite. So we can easily spot fakes using this technique.
John, who attends the Harrow International School in Hong Kong summarised his thoughts:
There is no trick that, when you use it, it will always figure out if a sequence is real or not. But there are a few things you can look for. By themselves, these are not helpful, but when you put them together, you just might be able to figure out if the sequence is real or not:
When there are extreme cases, such as 10 tails in a row, it's usually real because someone making a random squence won't include extreme cases as it doesn't seem random.
Real random sequences are not balanced at all. They usually have more of either tails or heads. If a sequence is very balanced, it might be fake.
People who make random sequences like to repeat ideas. If they have a lot of clusters (a few heads or tails together) at the start, then they are likely to carry on using clusters. If they have a lot of singles (an isolated head or tail), then they might carry on using a lot of singles. Random sequences are more varied.
Try putting yourself in the mind of someone who is making a sequence. Do you think that you could create something like that sequence?
People that make sequences are crafty. If a sequence meets all of the tips above, consider that it is fake, because although these are good things to look for, it is rare that they all apply.
And most importantly, don't let your thoughts limit you. The sequence randomly generated could exceed your wildest expectations.
Thank you to everyone who shared their thoughts about this fascinating problem.
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.