**Too Random, Or Not Random Enough: Student Misunderstandings About Probability In Coin Flipping**

On the one hand, simple experiments like dice rolling, coin flipping, and spinning spinners are very much overdone in elementary curricula around the world. On the other hand, they can be a way to experience and understand some truly massive ideas about chance.

Here is a lesson idea for coin flipping. I also wrote about this in my book, Teaching Mathematics Through Problem-Solving in K-12 Classrooms.

A coin flip is a true 50/50 proposition. It is, by definition, a fair game. We experience “50/50ness” as true randomness. If I flip a coin in front of you, and it lands heads, there is a decent chance that, if asked, you would predict that the next one would be tails.

So the idea for this lesson is this:

we think events are more random than they actually are.

Or put another way:

even in a probability experiment with just two outcomes, you will likely see longer streaks or strings of one outcome, given enough trials.

So here is the lesson idea:

- Have students make a fake simulation of a 100 coin flip experiment. They should produce on paper something that looks like this:

HTHTHTHTHHTTTHHHTTHTHTHTHTHTHHHHTTTHTHTHHTTTHTHHTT

HTTTHTHTHHTTTHHHTHTHTHTHTHTHTHHHHTTTHTHTHHTTTHTHTH

I have seen students write: HT HT HT HT HT all the way through to 100 outcomes!

2. Have them do a real coin flip of 100 trials, recording the results the same way.

3. The trials must not be marked real or fake, but students must know which is which. Now, you look at each pair of trials, and guess which one is real, and which one fake.

With all due rigamarole, announce that you are a mathematical magician, or some such thing, and that you can read their minds. Dramatize it as you like.

Try your own coin flip simulation here: Virtual Coin Flip Simulation. I will wager that you can do a 100 flip simulation right now, and you will not have a string of about less than 6 heads, or six tails, in a row. Even though the theoretical probability of stringing together a bunch of heads or tails is quite low, it is still 0.5 on each coin flip for either outcome.

Students understand what a 50/50 wager is, but their fake trials are TOO random.

HTHTHTHTHHTTTHHHTTHTHTHTHTHTHHHHTTTHTHTHHTTTHTHHTT

HTTTHTHTHHTTTHHHTHTHTHTHTHTHTHHHHTTTHTHTHHTTTHTHTH

When I look at this trial, there are only 2 strings of 4 heads, and none longer. I would have been pretty sure this one was the fake one. If the other trial had ANY strings longer than 4, I would have been pretty sure. One that stumped me had a string of 8 heads-that student clearly understands probability.

This is the principle of clumping. It explains why you can win the lottery two days in a row.

Here are some samples from a virtual coin flipper to provide some proof to my claim about “strings” of identical outcomes.

There is a string of 8 Tails in this sample.

There is also a string of 8 Tails in this sample.

6 Tails and 5 Heads strings in this one.

6 Heads and 6 Tails in this one.

Is this what randomness looks like to you? To students? Do this lesson and find out.

Need more proof? Start flipping!