# Everyday maths: Crossing the Road

### Here, let me get on this soapbox a moment

Things that irritate me beyond end, number two in a series of, well, lots: grumbles about how “oh, it’s pointless teaching maths, you only ever use it to work out discounts in the supermarket and there are calculators on your phone these days.” People asking “when will I ever use probability in real life?[1]”. The comment “algebra has served me no use since I learned it in school.” There’s a complete disconnect between what people think of maths and actual, proper everyday maths.

First up, if you show me someone working out the 3-for-2 discount in a supermarket on their mobile and getting it right, I’ll give you a free half-hour class[2]. And second, like a modern-day Billy Mack ruining an already-ruined Troggs song, maths is all around.

### I feel it in my fingers, I feel it in my toes

The original idea for this post was for me to go about my day as normal and list every maths concept that sprung up, from the fluid dynamics of the water coming out of the shower to the combinatorics of picking a matching pair of socks, to the seach strategies of finding where I’m up to in my bedtime reading, but quickly realised that this would turn into the Principia and, most likely, a secure institution.

So, I pared it down: what everyday maths goes into a simple task ? An everyday task like… crossing the road to get to the park with my morning coffee from the Little Red Roaster. Even this turns out to be a huge amount - and I’m certain I’ve missed out dozens more.

### The calculus of crossing the road

I stop at the kerb, like I was taught by the lollipop man when I was three, and look both ways for traffic. I see a car coming down the hill and make a judgement as to how fast it’s going, how it’s speeding up as it approaches me, and how long it’ll take to get here – a nice little exercise in integration and differentiation. Sure, I’m not writing anything down or consciously thinking of a speed-time graph – my brain, like everyone else’s, uses all manner of rules-of-thumb to do the maths. But it does the maths.

I also figure out a good direction to cross the road in – it’s a busy road, so I want to get across it as directly as possible. I pick a direction at right angles to the curving kerb: oh, look, it’s a normal! Did I figure out an equation of the curb, differentiate and do one-over-the-gradient? No, of course not. My brain did that for me.

### The statistics of crossing the road

There is always a danger of getting killed, injured or having your coffee spilt[3] when you cross the road. You minimise that risk, of course, by looking for traffic along the road – but you don’t waste energy looking into the sky for falling airships or into the park for runaway motorbikes: you know that the probability of traffic coming from those directions is tiny, so you don’t bother. Wait a minute, did you just make a decision based on probability? Here, have your Statistician badge.

Even by choosing to cross the road here, I’m making a statistical decision. I want to be in the park before my cappuccino becomes tepid and the froth becomes squishy (under the inexorable rules of Newton cooling and fluid dynamics, differential equations we also solve instinctively). I could walk 100 yards to the traffic lights, wait for them to change and walk back up the hill to my bench of choice; I have a good idea, to within a few seconds, of how long that’ll take and how cold my coffee will be afterwards. Alternatively, I can cross here. It could be a few seconds before an acceptable gap in the traffic opens up; it could be minutes and minutes. I may even end up going the traffic lights route or, worse yet, drinking my coffee on this side of the street. However, I’ve instinctively weighed up the probability distributions of my two main options, and decided that the risk of having to wait here is outweighed by the chance of getting to my coffee more quickly.

### The game theory of crossing the road

What the lollipop man didn’t teach me was that most cars don’t want to hit pedestrians. It doesn’t do anything for your insurance premiums or your mental health. Let’s say there’s a truck coming down the hill carrying just about 30,000 pounds of bananas. By the time it reaches me, there are four possible outcomes:

- All-round joy unconfined. The driver merrily waves me across, or I cross safely without concerning the truck at all. This is clearly a good option.
- I’m happy but the driver isn’t. There’s a slamming of the brakes, a thumping of the horn, a barrage of expletives from the cab, possibly a banana thrown in anger.
- The driver’s happy but I’m not. I’m tapping my feet on the kerb as he screeches up an extra 20 yards to the traffic lights, belching exhaust fumes in my face and ruining my coffee.
- We’re both unhappy. I’m in traction at the Poole Royal Infirmary and he’s being asked probing questions by the Dorset Constabulary about his speed at the time of the accident.

This looks a classical game theory problem – and if the driver doesn’t wave me across, I need to make a judgement about whether crossing would lead to outcome 2, which is perfectly acceptable to me, or outcome 4, which certainly isn’t. Again, I have to weigh probabilities, speeds, the dynamics of banana-shaped projectiles, all kinds of mathematical concepts and make a decision. And again, I don’t draw out probability trees or a minimax table, my brain makes a judgement call – but it’s doing everyday maths, whether you like it or not.

### This soapbox is getting uncomfortable

I shall use the principles of mechanics to descend safely from it, and some very clever mathematical modelling from the Alessi company to make myself a cheeky lunchtime espresso. This (as we know because of a great deal of statistical experimentation) will give me a temporary boost in mental capacity, meaning I can reap the benefits of information theory and computer science to give a class over Skype this afternoon.

So much for a pointless subject with no real-world applications!

[1] Actually, it’s not the question that bugs me. It’s a perfectly valid question. It’s when it’s used in the sense of “I can’t be bothered remembering how to do this sum, it’s sunny outside and I have a tan to work on” that bugs me. [2] Although, I never go to supermarkets these days, so you’ll have a job. [3] I guess unless you cross the road without coffee. But who would do such a thing?