Monday, 22 June 2015

Counter-Intuitive Probability: Monty Hall Problem

The Monty Hall Problem

This is one of my favourite maths enquiries.  The children find it fascinating and because it is counter-intuitive, it really makes us think about our central idea:

'Instead of luck, mathematicians think of probability'  

It's taken me quite a few years of relooking at it to finally fully understand and break my intuitive stance on it.

This Youtube explains the Monty Hall problem:


Lead in:


We began by discussing the word 'intuition', what we felt it meant and examples in our life when we experience it.  Some really fascinating theories emerged from this as we discussed where intuition comes from!

We then discussed what it might mean when we say something is counter-intuitive and examples in our lives.

What Counter-Intuitive Probability Looks Like:

A child came out onto our 'TV game show'.  To win the car, they needed to choose a door: A, B or C.

The child chose B.

I asked everyone to think to themselves what probability the child has of winning.  Think of it as fraction, what is is as a %, and what is it as a decimal.

I revealed one of the other doors that had a goat behind it and gave them the option to change doors. 

As the student was thinking about what to do, I asked the others to think to themselves what the probability is now of winning and if there it made a difference by changing doors or not.
The student decided not to change doors.

Like most of us, our intuition tells us not to change doors.

I asked what chance we thought we had now of winning the car.

We figured we have a 50-50 chance (it's not, but we didn't know that yet).

We tend to stick to our original choice because we would be quite upset with ourselves if we did change and the car was behind our original choice. And because we believe we have a 50-50 chance anyway, it would be foolish to change.

To try to sway some thinking, we did the game several times with different students (and with me changing where the car and goats were).  Annoyingly for me, the kids quite often chose the car behind their first choice (thanks probability!) and wouldn't change when given the option so I kept inviting more students up to play till it started sparking some interesting comments.

I gave them some labelled cups and images of goats and cars. In partners they openly experimented with the game for 5 minutes.

We shared some discoveries we had made.

- The probability of winning gets bigger with the second choice because it grows from a 33.3% chance to a 50% chance.   (I knew this wasn't true, but know it's better for them to discover this out for themselves)

- We seem to win more when we don't change doors (Oh dear! Is this going to go pear-shaped?!?)

- Not for us! We win more when we change!!   (Phew!!!)

The class was then split in half.

Half the class tested the game this way:   Every time the 'contestant' partner played, he /she could not change cups when given the option.

The other half of the class's 'contestant' had to change each time.

They needed to design a way to record the results.

I explained that when we have finished conducting our experiment to see if there was a difference, we would convert the results to a % so how many times should we do the test?

We agreed upon 20 times as 20 can be easily converted to a %.

We gathered our results together by creating a % of how many times their 'contestant' won the car.

Someone suggested we should find the average. Wow! Great idea!

I asked what we noticed about the results:

- We have a better chance of winning if we change.

- But why?!? That doesn't make sense!!

- It must have something to do with probability.

- But on the second choice, it is only a 50-50 chance. These results are just showing good or bad luck.

- But remember our central idea? Mathematicians don't believe in luck, they think about probability. (Love it!!)

- So why is there a higher chance of winning when we change doors?

That last one, was the question I was waiting for.  We then visually drew what we thought the probability looked like.  Most of us drew tree diagrams, but got muddled by the maths with that second choice as expected. 

Together we then pulled our thoughts together and had our ideas written on the board.  Some of us had shed the idea of the second choice being a 50-50 chance and led us to think whether it had actually became a 66.6% chance when we decide to change.  

This was when some more light bulbs started flashing.

- Ohhhhhhhh   Now I get it!!!

- It ISN'T 50-50, but my gut feeling says it is.

- You have to connect the second choice with the first choice!!  (Big light bulb moment!)  You can't think the second choice is independent of the first choice! 

- Oh yeah! If you think you are playing two separate games you go against the real probability. The two choices ARE connected!!

Someone suggested we should try a different experiment with a new hypothesis.  We agreed that was a great idea so that's what we did. 

We had all different sorts of hypotheses:

° If you change doors, you have a higher chance of winning. 

° If you don't change on your second choice, you have a lower probability of winning. 

Some more sophisticated hypotheses:

° If we choose not to change doors, we have a higher chance of not winning because we only have a 33.3% probability.

° If we choose to change doors, we have a higher chance of winning because we then have a 66.6% probability.

We tested our hypotheses and shared the results from our experiments; we found all of our hypotheses were supported by the results!

We had discovered that we do increase our chance of winning by changing doors.

We begin the game with a 33.3% of winning.

When given a choice to change and a door is revealed, if we stick to our original door choice, we stay at a 33.3% chance.

But, if we change doors, our probability of winning has just increased to a 66.6% chance.   We actually double our chances of winning by changing doors.

We noted that it does not make it a guarantee to win, but if played several times, then our probability of winning increases.


On post it notes, we reflected on how this activity helps us with our understanding of our central idea:

'Instead of luck, mathematicians think of probability'

We shared our reflections in small groups and a few shared with the whole class.

Taking our learning beyond our classroom:

Someone asked if we could take the cups, goats and car images home to see what their parents would do and then teach them the maths behind it.  Lots of us also wanted to so they took home the cups and images of the goats and car.  

I told them I had found a TV show video on YouTube that tests out the Monty Hall problem in a similar way that we did and if they'd like me to email to them to watch at home tonight.  With great enthusiasm from most, they did.

The Youtube link:

There is that barometer again:  Surely we are doing something great in maths when the children want to take it beyond the classroom. :)

To sum up the learning, I asked them if they are ever on a TV game show and play this game, what should they do.  Everyone called out: "Change your Choice!"

And after you win the car?

- Find Mr Anshaw and thank him!

Exactly.   I asked them all to promise if they do win a car from a game like this, I'd appreciate a selfie of them driving their flash new car. 

They all promised they would. :)

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