## A probability puzzle

*“Colin and Dave are playing a game. Colin has a probability of 0.2 of hitting the target with any given shot; Dave has a probability of 0.3. Whoever hits the target first, wins. Colin goes first; what is his probability of winning?”*

Yesterday I listened to the latest edition of “Wrong, but useful” (@wrongbutuseful), and the above is Is the puzzle set by the cohost Dave “The king of stats” Gale (@reflectivemaths).

It was pretty late on in the evening, but I decided to have a quick attempt at the puzzle nonetheless.

Here you can see my back of the envelope workings, complete with the word “frustum” on the top of the envelope as I had added an extra r into it in my recent post. You will also see that approaching midnight on a Saturday after a day at the NTEN-RESEARCHED-YORK conference is not the most idea time for solving maths puzzles! My thinking was fairly valid, I think, but my tired brain has made an absolute ton of mistakes! *(If you haven’t spotted them, go and have a look before reading on!)*

When I looked at it this morning the first thing that jumped out at me was “0.14×0.028 does NOT equal 0.0364. Then I thought, “ffs, 10/43 is very definitely not 0.43!” This was closely followed by: “and why have you used 0.14 as r, the 0.2 is the probability he hits, if he hits the game is over, Cav you ploker!”

So I had another go:

I think this is right now. (Although do feel free to correct me if you spot another error!) The probability Colin wins on the first go is 0.2, on his second go is the product of him missing (0.8) Dave missing (0.7) then him hitting (0.2) so 0.8×0.7×0.2. As the sequence goes on you are multiplying by 0.7 and 0.8 each time, so it gives rise to a geometric sequence with first term (a) as 0.2 and common ratio (r) as 0.56.

The total probably of Colin winning is the sum of the probabilities of him winning each time. This is because the probability he wins his the probability he wins on his first go OR his second go, OR his third etc. The game goes on until someone wins, so is potentially infinite, thus we need to sum the series infinitely.

As the series has a common ratio of 0.56, which is less than one, we can sum the series to infinity using s = a/(1-r) which gives 0.2/(1-0.56) = 0.2/0.44 = 5/11. Thus the probability Colin wins is 5/11 or 0.45recurring.

## Show that questions

On the recent episode of “Wrong, but useful” Will Davies (@notonlyahatrack) asked the hosts Dave (@reflectivemaths) and Colin (@icecolbeveridge) and guests Samuel (@samuel_hansen) and Peter (@peterrowlett) to discuss ‘why do students hate show that questions?’

I was was a little surprised by the question, as my students love them. They get excited by them because they know where they’re headed and if they make a silly mistake it means they haven’t got the right answer and know to check through to find the mistake (often with my year 13s it’s something really silly because they’ve rushed, such as a plus sign becoming a minus sign or a daft arithmetical error such as 2×3=5!) (nb that’s an exclamation mark, not a factorial sign).

The discussion on the podcast was interesting, but it was accepted by all four members of “the panel” that students do hate them. It really got me thinking, are my students weird? Is it something to do with the way I teach them? I know my students tend to love the subjects I love and, as mentioned here, have often picked up my preferences for methods. Have I drilled my students in such a way that I’ve made them love these questions? Perhaps I have, but I think that’s a good thing. They are the questions where you can be sure to eliminate silly errors, so if you have the mathematical ability, you should have the marks.

I was fascinated by the discussion though, Dave suggested that the hatred could be built from the myth that maths has a right or wrong answer. This is a myth that I find irksome. It’s a total fallacy and yet people who lack knowledge about maths assume this to be the case. We’ve all had those comments “ha, marking? You’ve got it easy, just tick or cross right?!” These negative perceptions of maths are often passed on by parents, and sometimes by teachers of other subjects! I think this could be due to their own experiences of the subject. This is something that needs to change, Maths is about a LOT more than right or wrong, as maths teachers we need to work on this. I’ve touched on this before here.

During the discussion Dave also suggested that, as we are interested in the mathematics more than the correct answer, perhaps we should have maths tests where the correct answers actually have no marks attached, where all the marks are attached to working. This is something that I find intriguing. It can be infuriating to have to give full marks to an answer with no working (it could be a guess!) but give another pupil only half the marks despite perfect working because they copied a number wrong from one line to the next. I would, however, worry about how the marks are awarded. As we know, there can be many equally correct ways to get to the right answer, and all would need to be able to get the same marks, otherwise we wouldn’t be testing mathematical ability, rather we’d be testing the ability to guess what’s on the markschemes!

Do you have a view about Dave’s proposal? Do your students like those show that questions? I’d love to hear your views/experiences.

## Maths and Numeracy GCSEs

A while ago I read this piece from @bigkid4 about his vision for the future of maths exams. It’s a great piece, with plenty of food for thought, and I’ve thought a lot about it since. Then the other day I was listening to Colin Beveridge (@icecolbeveridge) and Dave Gale (@reflectivemaths) chatting on their podcast “Wrong But Useful“, about the need (or lack of) for all pupils to study maths to GCSE level. Colin suggested a similar idea to @Bigkid4’s blogpost, which was to split the subject into two qualifications. A “Numeracy” one and a “Maths” one, making the Numeracy compulsory and the Maths optional. (nb I have no idea is Colin read this blogpost, but I know it’s an idea I, and others, had had prior to reading it. Great minds and all that…)

Colin and Dave had a long chat about this on the podcast (which was excellent as always- if you aren’t a listener, but like maths, then become one! You won’t regret it!). They postulated that the numeracy qualification was perhaps not big enough to be equivalent to a GCSE and they had a few nice ideas about what could be on it.

That got me thinking,

if we did split it, what should go where? And could the numeracy alone be a GCSE?I think that we could create a stand alone GCSE, but I would call it “Numeracy and Problem Solving”, I would keep the subjects together until KS4, ensuring a basic grounding in each. I would like to see plenty of number on the NPS syllabus, including Fractions, Decimals and Percentages. Certainly things like interest, compound interest etc are paramount in a world of unscrupulous bankers and payday loan sharks. I would like a massive part of it to be around estimation and checking (and, of course, rounding!) include a focus on the all too often missing “common sense“! I would have some basic probability on it to ensure pupils know how daft some gambling is. I would like to see some modelling on there too, as I feel that is a neglected skill at KS4, and one that is extremely important. There would be a large focus on problem solving, and I think it would be imperative to include a bit of basic RAT Trig (Right Angled Triangle Trigonometry. I would say “and Pythagoras’s Theorem”, but that would anger Colin, so I will say “including Pythagoras’s Theorem,” instead). I guess it might seem strange at first, but I know that it’s a skill joiners use, and others in trades that pupils who I know wouldn’t choose maths have ideas to go into.

I also think that charts etc should go here (although boxplots, stem and leaf and any of the other nonsensical ones should be scrapped and forgotten about forever, along with transformations by hand!)

That would leave maths. The higher number stuff would be here (prime factorising, etc). The majority of GCSE algebra would be there too, and some a level stuff too (I’d forget about the GCSE nth term stuff, and go straight to the A level stuff!) I’d like an introduction to calculus and an introduction to topology. Some higher probability stuff, and a good basis to build A Level stats on (variance etc). Perhaps and introduction to matrices and complex numbers too!

I think there are many positives about these ideas, but I would worry about the take up of maths GCSE, and the potential damage that could do to maths A Level and undergraduate take up further down the line.

If you have anything to add, please comment, and if you have any links to other blogs etc around these ideas, I’d love to have them too.

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