Brilliant Bounds!

In a previous post I wrote about fractions that appear on roadsigns. 

I had gone with an initial assumption that quarters and halves were the fractions that appear on UK road-signs, but then we saw one that featured one-third!   

My new assumption is that these fractions appear on UK road signs:

 

This is brilliant if we want to think about upper/lower bounds.

Assuming that the bounds are halfway between one measurement and the next, this gives:

If we put them all over 24 then the upper/lower bounds are:

 

There is something rather pleasing about these!

Least accurate measurement?

According to the Department for Transport: “A sign may indicate the distance to a destination in miles. Fractions of a mile may be shown for distances less than 3 miles.”  This means that distances less than 3 miles can be given rather accurately because they can use the fractions shown above. 

A final question: which exact measurement (whether greater or smaller than 3 miles) has the largest percentage error?

As I was going to St Ives 1: See the signs

No - not the old riddle, even though it appears in DieHard: with a Vengeance and would link with an earlier blog (and it probably isn’t that St Ives anyway - I was in Cambridgeshire).

[Incidentally, I never quite understood why I couldn’t be going in the same direction as the polygamist - surely all those “kits, cats, sacks, wives” would slow a man down so I could easily catch him up?  2802 seems a perfectly sensible answer to me (although it assumes that I am not carrying sacks, cats, etc too).]

I took these photographs on a bike ride close to St Ives.

What maths can we do with a class with these?  Lots, it turns out.  Let’s start with just the first picture (but before you do you might want to ask pupils what is strange about the way fractions are written on road signs - most of them will never have spotted that they are written without a line):

• How far is St Ives from The River?  
• What problems might there be with that answer?   There might be a more direct route from St Ives to The River that is shorter than 2¼ miles.
• What about the accuracy of the measurements?  Presumably they have been rounded
• What degree of accuracy might have been used?  
• So: what is the maximum distance of The River from St Ives?  From this sign the most precise measurement involves quarters, so maybe everything has been rounded to the nearest quarter.  In this case the biggest River distance would be 5/8 and the biggest St Ives distance would be 1 7/8, giving 2½ miles as the maximum.

Now let’s put picture 2 alongside picture 1:

What can we do with this one?  I would let pupils decide what questions they can answer.  Ideas include:

• How far from St Ives to Fenstanton?
• How far from St Ives to Hilton?
• How far from Hilton to Fenstanton?
• Why might these answers not be accurate?
• What if we consider the rounding that has taken place? 
• What happens if a lorry needs to go to Fenstanton?
• Combining it with the first picture: what is the maximum distance of Hilton from The River? 
• What is the minimum distance?

Finally, let’s add picture 3:

We could repeat similar sorts of questions with this one, including: “How far did I cycle between taking pictures 2 and 3?”. 

But hold on: on this one it says that Fenstanton is 2 1/3 miles away.  Our previous assumption about rounding to the nearest quarter is incorrect! 

So, what is the upper bound of signpost measurement given as ¼ ?

More on this in a future post!

Binary trading is bad

Some of my mechanics-studying sixth-formers were discussing Binary Options yesterday.  This worried me a little because I have always taken it for granted that Mechanics was a superior discipline to Statistics and their scary lack of statistical knowledge has made me start to reconsider this!

When I described it as ‘gambling’ they corrected me: “it’s investing”.

The way it works is that you bet (sorry, ‘invest’) that over a particular time-frame (that might be only 5 minutes) a particular share or basket of shares will rise or fall in value.  That’s it.  Now, using your superior knowledge of the markets to decide that over a period of several days/weeks/months/years a particular company will rise or fall in value is one thing, but to get through the ‘noise’ of minute-by-minute share-dealing to say that a share will rise/fall immediately is very unlikely (unless Sir Alex has given you the tip that he is about to announce his resignation!).

Forbes business magazine puts it like this: http://www.forbes.com/sites/investor/2010/07/27/dont-gamble-on-binary-options/

Quoting that article:  One particular binary option website

pays $71 for each successful $100 “trade.” If you lose, you get back $15. Let’s say you make 1,000 “trades” and win 545 of them. Your profit is $38,695. But your 455 losses will cost you $38,675. In other words, you must win 54.5% of the time just to break even.

How did Forbes know that it would be 54.5% of the time?  I reckon my non-stats sixth-formers (and Higher tier GCSE pupils) can work this out for themselves.

Let’s call the probability you win p.

In the long run you will win 71xp dollars per ‘trade’.

The probability you will lose is (1 - p) and in the long run you will lose 85 x (1 - p) dollars.

To break even your wins and losses must be equal, so we have the equation:

71p = 85(1 - p)

Solving this gives p = 85/156

Which is about 0.54487

How to present this to a class?

Slide 1:

There is a particular type of gambling that is close to being random.  You pay $100 and if you win you get your stake back and also $71 in winnings.  If you lose you get $15 back.  Should you play?

Slide 2:   (Scaffolding)

Let’s call the probability that you will win each time p. 

How much will you win each game on average?

What is the probability you will lose?

How much will you lose each game on average?

What probability do you need so you will break even in the long run?

Slide 3:  Why do you get $15 back even if you lose?  What is the point of starting with $100 in that case?

I will try it out with a KS4 Higher class in a couple of week’s time.