## 100% Chance

*“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.

## A quirk of probability

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!

## All you need is sine

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 hypotenuse 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.*

## Decending powers of x

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!

## Bizarre multiplication

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!*

## Mathematical Graffiti

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!*

## Statistical Deception

When teaching and talking about statistics I always emphasise the need to be careful what you believe and to always ask yourself “what agenda does the person presenting this data have?”

I’ve written before about how stats can legitimately be manipulated to serve different points of views, especially when there are false variables at work. But recently I’ve noticed at darker art in statistical manipulation, one that is, at its heart, lying.

We are less than six weeks away from local elections now, and it is becoming silly season for party political leaflets coming through our letterboxes. Now we all know that the political parties will present data in a way that makes them look better, they are trying to win your vote afterall, but we would expect them not to lie. For the data to be accurate and presented correctly. Unfortunately, however, this is not always the case:

**Exhibit A**

This popped up a number of times in my twitter feed from a variety of sources. I believe it is from a Lib Dem leaflet in Manchester. As you can see, they have presented a bar chart with proportions labelled as percentages. The first screaming error is that the red bar and the orange bar are massively different heights, yet are both emblazoned by the label 39%. The second glaring error is that the percentages add up to more than 100%. The first implies that either the Lib Dems are deliberately trying to mislead voters into thinking they are in a stronger position in the ward than they are, or that they don’t realise that 39% is equal to 39%. I’m not sure which is worse?!

Here’s an excel interpretation of what the graphs **should** look like:

**Exhibit B**

This graph came through my door in Leeds North West parliamentary constituency. The first thing that caught my eye was that although the gap between the number of votes between Lib Dems and Labour; and between Labour and Conservative is almost the same, the difference in the gaps between the bars was almost 5 times as big, which would imply almost five times as many less votes! An obvious fallacy. Either it’s a deliberate attempt to mislead, or they can’t draw a bar chart. If it’s the latter, do we want them in charge of our local authority budgets?! (or the entire economy for that matter!!)

Something else that struck me as deciving, although this time mathematically correct at least, was the choice of data. This was a leaflet issued in the run up to a local election, and the data set used was from the last local election. Why then, is the data that for the parliamentary constituency rather than the council ward? The ward makes up around a quarter of the constituency, and the vote share in the ward is radically different to that of the constituency. The sitting councilor is conservative and sits on a huge majority, and the Lib Dem candidate last time out cane third. To issue a leaflet in the run up to a local election which implies the conservatives can’t win in a ward where they have a large majority and back it up with local election data for a parliamentary constituency is deliberately deceptive and misleading.

Here’s an excel interpretation of what this one **should** look like:

**Exhibit C**

This one comes from *“across the pond”* and is another which was viral. This one seemed to appear constantly for a few days everywhere I looked. If you are still wondering what’s wrong with it, take a little look at those numbers down the left hand side…. See it? The y axis goes upwards to zero! Drew Barker (@twentythree) made this version which gives a much better picture as to what’s going on.

I can’t wait to see what my classes make of these!

*nb I haven’t “selected” these graphs as an attack on the Lib Dems, it’s just they are the only party who have sent me a leaflet with incorrect maths. I’ll gladly expose any of the parties if they themselves do. I do collect these, so if you spot anything similar, do send me it!*

## Abacii and Abbey House

Quite a while ago I wrote this post about Abacii. I was hoping to buy one for my daughter, but everyone I found only had ten beads per bar, and as we live in a base ten society this makes no sense, and we should have nine beads per bar. This week we took her to the abbey house museum and we found this:

The writing next to it clearly states that you use an absence of beads for zero, each line being for a different “place” in the number. And that when you get to ten you move one bead on the row above. This would be fine, if there were 9 beads per row, BUT THERE ARE TEN!!!!!! If on the tenth bead you swipe them all back to the other side and move one on the next row then the tenth bead should never move, it is obsolete, why is it there?!?!?!? I still haven’t found one for my daughter with nine beads on it and I can’t fathom why they all have ten. Its entirely ridiculous.

On a brighter note, they had more maths references in the museum. They had this:

Is on the bottom of a black board and has helpful workings to show how to add, subtract, multiply and divide. The implication was that at some point in history these were common in classrooms, and I thought they were nice.

There was also this little poem:

The negative-ness of it shows that maths has always had a bad press! This sort of feeling is still rife, and we are constantly battling to destroy it.

## How many ginger nuts make a serving?

Biscuits have become more prevalent during lockdown in my house. Snacking in general has. This is probably no good thing but its certainly enjoyable. Today’s snack of choice was a brand new packet of Ginger Nuts:

25p from ASDA, a steal if ever I saw one. When I looked at the packet I noticed this:

“We suggest this product provides 25 servings”. I thought, “I wonder how many biscuits they suggest for a serving?”

How many do you reckon?

I did a bit of mental estimation, it didn’t look a big enough packet to contain 50 ginger nuts, so I assumed it would be 25 biscuits. Then I counted them.

It wasn’t 25 biscuits, it was 26! (n.b. that’s an exclamation mark there, there wasn’t 403291461126605635584000000 ginger nuts contained within.)

That’s 1 and 1/25 (or 1.04) biscuits per serving! What a bizarre amount. Who are these people cutting their 26th ginger nut into 25 equal slices? How would you even go about doing that?!?!?!

If you’re one of these people, please do let me know!I mentioned earlier the price. 25p per pack. That’s 1p per serving, what a delight. 1p per 1.04 biscuits! Or 0.96154 biscuits per penny! What value.

Note: ASDA have not paid for or sponsored this post, but I’m happy for them (or anyone else for that matter) to send me free biscuits if they wantany more reviewing.## Share this via:

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