Really? 4 Dice Can be Beaten by 2 Dice?!?!?

Lead In:

We began by thinking of three different ways we could explain our central idea:

'Instead of luck, mathematicians think of probability'

- Probability can be measured, but luck cannot

- We can use fractions /decimals or % when measuring probability, but we can't do that with luck precisely. 

- We can have different perspectives when we look at a situation, we can see something as being a lucky event or we can see it mathematically.


Brilliant. :)




We thought that last idea was really interesting and there were some confused faces so we tested out how we can do that.

I asked them how they could see what they had for breakfast this morning from two different perspectives:

- a luck / lack of luck perspective depending on what you ate

OR

- a mathematical probability perspective


Some confused and some wondering looks.

Let's look at our breakfast this morning wearing 'luck' glasses.  How could we see it that way?

- We are really lucky to have food compared to other people in the world.

Absolutely.

- This morning my Dad made us pancakes.  I thought today was my lucky day.

etc

We then took off our 'luck' glasses and tried to see our breakfast wearing 'probability' glasses. 

It took us a while to think this way, but then a student mentioned she has 1 in 7 chance of eating toast. Every day she must eat cereal, but once a week she is allowed to have toast. After this first idea, loads more were generated.  Some of us shared how many different types of cereals they had in the cupboards so they then could work out the probability of being served a specific cereal. 

- Wow! Maths really IS everywhere.

It sure is.


_______________________________________________________

I found this great probability problem on YouTube which I thought would help us with our new understanding of intuition and how maths can sometimes prove that wrong.

YouTube Link:The Last Banana Probability

We stopped just after the problem was explained and retold the situation to our table partner to ensure we all understood it.

Basically, we have to choose whether it is better to be Player 1 or Player 2.

They roll 2 dice.

If the highest number rolled is a 1, 2, 3 or 4 then Player 1 will win.

If the highest number rolled is 5 or 6 then Player 2 will win.

When asked to think about which player had the higher probability of winning, nearly all of us thought Player 1 did (see image).

What could we do to find out whether our idea is correct?

- We could conduct an experiment by rolling two dice 20 times and see if the results support our idea.

- We could create a hypothesis and test it out.

- We could work out all the possible combinations and then see which had the higher probability.  (This is the better idea, but I wanted the children to learn this for themselves, so they were to explore this with a partner and report back their findings in 10/15 minutes)

These students were surprised to find how much more Player 2 was winning.


But WHY?!?

These students had cracked it!  By recording all the possible outcomes first, they then created a fractional probability.  They figured Player 1 had a 4/ 9 chance of winning compared to a 5/9 chance for Player 2. 

                                                                                                                     Brilliant!

I was curious how these students had come up with the probability for Player 2.   They hadn't actually considered the possible outcomes.  They had done an experiment and used the results to try to formulate a probability. 

If you did this again, would you expect the same probability?

- No, it would be different.

                                                                                                              Does that make sense then?  

                                                            - Oh wait! I know what we need to 
                                                               do!!!

These students explained they had conducted an experiment of who would win and then they converted the results to a % to make it easier to understand.

Great thinking.

But, I knew it wasn't showing the right probability.





Those conducting the experiments weren't really getting to the nuts and bolts of the probability.   So, all the partners on one side of the room had 1 minute to go to the partners on the other side and find out what approach they were taking with their investigation.  Then the other side of the room did the same.  I was hoping that by doing this, it would spark some new ideas of how they could investigate more effectively.  There were enough in the room working out all the possible outcomes first and later experimenting to allow this better idea to start filtering around the groups some more.




After some more time on this, we then grouped together and shared what we had discovered.


Most of us were really surprised to discover that Player 2 had the higher probability of winning (56% chance) even though they only had 2 dice numbers to win on.


Those amongst us who had cracked it shared the tables they had created that showed all the possible outcomes.


Light bulbs started flashing in the discussion and so then we finished watching the YouTube to get a better visual understanding of the probability.

Looking at the table on the YouTube, we could more easily see how 5 and 6 did actually have a much higher probability even though it didn't make sense to us in the beginning.

Someone mentioned intuition and our gut feeling from yesterday and we wondered if that was why we first thought Player 1 had higher odds of winning or whether it was more a quick mathematical estimate. 



Reflection:

We did an oral reflection to help reinforce what we had learnt with a different table partner by using the 5 Whys strategy, but we reduced it to 3 whys.



It sounded a bit like this:

1. Why do we need to work out the possible outcomes to find the probability of a situation?

- Because if we don't, we can't see which has a higher probability.

2. Why?
- Because you need to know the total amount. 

3. Why?
- It's like with percentages.  A percent is out of 100.  The only reason you know if something is a high or low percent is because you know it is out of 100.  It's the same when measuring probability.  You need to know the total amount of possible outcomes to then be able to determine if more of them will let you win at something or not.





Actually, this was one of the better '3 Whys' I listened into.  Others struggled with it a fair bit, so I have to think more before using this strategy again in the future as a reflective tool.   Maybe I just need to give my students more opportunities at using it (we seldom do) in order to be more proficient at it (?)


Something for me to think about.......