“100% Chance of getting a safety car.”
That line was repeated numerous times in the build up to today’s signapore grand prix and in the early stages of the race. It was repeated by various pundits and comentators and it has made my blog boil.
THERE IS NOT A 100% CHANCE OF A SAFETY CAR
That should be enough, to be honest. It’s not certain that it will happen, there is a chance that it might be avoided. The problem stems from the confusion between relative frequency and probability.
Relative frequency IS a good proxy when it comes to probability but it’s isn’t always exactly the same thing. The relative frequently of a safety car being needed at Singapore was, still is, 100% because There has been one at every race ever held here, which gives us a relative frequency of 100%. But the sample size is tiny (9 races) and this isn’t big enough in this case. Please sky sports, sort your maths out.
Today I was playing a game on my phone. Marvel puzzle quest if you’re interested. It’s based on characters from Marvel, follows story lines and it’s game play is a bejewelled type match game. If you enjoy those games and like Marvel then it’s worth a download. And let’s face it, you’re the sort of person who spends their free time reading maths blogs. So there’s a good chance you will enjoy it!
Anyway, back to the point, at the end of each level you recieved a reward, most levels have 4 possible rewards and you get one of them chosen at random. You can replay the level and earn all the rewards, each time you get one at random, however the pool does not decrease and if you are randomly assigned one of the ones you have already won you get a non reward, as it were.
This morning I had completed a level. I’d beaten Venom, for those of you interested. I then replayed it to try and gain the other rewards, after 6 goes I’d had the first reward then 5 non rewards. What are the chances of that? I thought, then quickly answered 1/4^6 obviously. That’s all well and good, but then I thought “it’s got to be a different one this time!” And got very angry at myself.
These are independent events. The probability of the next one being a non – reward is clearly still 1/4. Yes, the odd of getting 7 in a row are tiny (1/4^7) but so are the odds of getting 6 in a row then one of the others (3/4^7). This is something I spend a lot of time discussing in lessons, because I know it’s something learners often have trouble with, so I should have known better than to let that thought sneak into my mind.
I was reminded of the Derren Brown show where he threw ten tails in a row on a fair coin. It’s a clip I often use in lessons and ask students to conjecture how he did it, rarely do any get the right answer (I think perhaps only once!) The answer, for those wondering, is that he threw the coin over and over until he got ten in a row, every time he got a head he started the count again. I think he said it took him 16 hours. He could do this because he knew it would probably happen eventually.
In both these instances, this row of events taking place looks highly unlikely. However Derren repeatedly did it until he got the result he was looking for and I’m probably one of millions of people playing that game and so it’s bound to happen to someone.
Probability is a funny thing, and I think a bit of knowledge about it can lead you to fall into the trap I did. I knew the probability of 7 in a row was low, but for a second forgot that we were looking at the probability of 7 in a row given that we already had 6 in a row!
Today I was going through an M1 question with a year 13 student and was surprised to see the method he had used. The question involved finding an angle in a right angled triangle given the opposite and adjacent sides. The learner had used Pythagoras’s Theorem to find the hypotenuse then used the sine ratio to find the angle.
Puzzled I questioned further, thinking he may have instinctively found the hypotenuse without fully reading the question then having all 3 sides so going with the first. This turned out not to be the case:
“I know sine equals opposite over adjacent innit sir, I have trouble remembering the other ones so I just always use sine.”
This was extra interesting as earlier I had come across a markscheme which suggested the way to resolve a force at an angle of 30 degrees was to use Fsin30 for the vertical and Fsin60 for the horizontal! Further checking showed this learner did that too.
I wasn’t too sure what to make of it. It’s mathematically correct, so there’s no issue there. The learner has a grasp of the other ratios but is more confident with sine so I can see why he would default to that position, although I hope the extra time it takes isn’t an issue tomorrow. I can’t fathom, however, why the markscheme would show it this way in the first instance. (Not the only time a markscheme has confused me recently!)
What do you think? Have you got any quirky methods like this? Have any if your students? Do you have an idea why a markscheme would default to this position? I’d love to hear your response.
I was in one of my colleagues lessons this week.and he was teaching the class to expand quadratic brackets. As the lesson went on he noticed that a number of pupils had been writing the X squared term, then the constant term then the X term so he pulled the class together to tell them that conventionally we write quadratic equations in decending powers of x. This is excellent practice and something we all should be encouraging, but it made me think “Why decending powers of x?”
When dealing with quadratic, cubics and quartics up to GCSE and A-Level level we use the convention of decending powers. This is common in all sorts from expanding through long division, even in linear function we use descending powers. However, when we start with series expansions, such as binomial or Taylor’s, we switch to ascending powers. I’m also fairly certain that everything in my second and third year university modules on polynomials was in ascending powers too.
Now don’t get me wrong, I’m happy with the fact we have different conventions here, and I’m not against using them. I’m just inquisitive, and would love to know if there is a reason, and if there is what that reason is. If you do know, I’d love to hear it!
A while ago I wrote this post about multiplication. In it I explored a few different ways of multiplying and what I felt worked best and why. I covered the main ones, and I realise there are many different methods.
However, today, while marking my year 8 books I discovered a method I had never seen before:
I looked at it and thought, “bizarre”. I asked the girl in question who taught her it and she said she’d didn’t know, but that it made sense to her but grid and column methods don’t.
I like the method, it reminds me of expanding a pair of binomial brackets, but it the numbers were 19 + 4 etc. It’s fantastic for 2 digit by 2 digital or 2 digit by 3 digit and I think it shows the distributive property quite well. There are major limitations though. It looks really neat in these two examples, but the questions that were 3 digit by 3 digit were a little spaghetti like and the ones with 4 digits were totally illegible.
Have you seen this method before? Do you like it? I’d love to know!
Pi appears in the strangest of places. It is, of course, the ratio of circumference to diameter for all circles. It is also half the period of the sine cure and cosine curve, and hence a sinusoidal wave. It’s in the period of a pendulum, heisenberg’s uncertainty principle, cosmology, Fluid Dynamics and many other maths/physics formulae. It crops up in nature in the double helix, it’s even the mean meandering ratio. (The ratio of a rivers actual length, to the straight line distance from source to end.)
This, however, seems to be ridiculous:
I spotted this piece of graffiti on a toilet door at a wedding, and before you ask no I didn’t put it there.
I did wonder if it could be an accident, could some of the paint have peeled off and just happened to form this Shape? Perhaps, but there was hardly any other paint missing and the shape was so perfect I decided it had to be deliberate.
So, is there a rogue mathematician tagging the toilets of Yorkshire with his favourite constants, or was it a Greek person who was caught in the act after just 1 letter?
Have you encountered, or committed any mathematical graffiti? I’d love to hear about it,and see it if you have!