## Wednesday, October 03, 2007

### Making The Right Choice

Everyday we are confronted with numerous occasions which call for decision-making, and quite often the decisions are related to numbers.

For example, given two options of sugar in a supermarket,

• A one-kilogram package which costs one dollar,
• A five-kilogram package which costs four dollars.

Some people reckon that the latter is the better choice, since it is "just eighty cents per kilo". However, some others might say that the former is better because you do not risk the sugar expiring before you finish it, the quality might deteriorate if it's left for too long, "you do not need five kilograms anyway", and so on. Well, there's no clear winning side here, as each pros and cons are weighed differently by different people.

On another occasion, a similar dilemma arose in the issue of tram tickets. As a background, there are a few choices for the prepaid tram tickets, namely two-hourly tickets, daily, and 10 X two-hour. Daily is out of the question because it could be replaced by an equivalent of two two-hourly tickets. So We could buy separate two-hourly tickets (which cost 3.30 dollar each) or buy a ticket with 10 two-hourly values (which costs 27.60 dollar).

Naturally, 10 X two-hourly ticket seems to be the best option. But here comes the real problem - how many of them should we buy? We know very well that there is a price increase every year or two, so if we stock up the tickets we would not need to buy expensive tickets in the future. However, buying lots of tickets turned out not to be the best choice after all - we figured out that at the present rate of bank interest and the predicted price increase every year, we might as well buy the ticket after the price increase, because by that time the bank interest we earn would have covered the price difference. It's quite a bit of tricky maths.

The last case I would like to talk about here is related to medicine. You might think, hey, what does medicine have to do with mathematics? Is it the doctor's bills? Hah let me reassure you that billing is not the main use of maths in medicine. As I mentioned in another post, a lot of medical researches deal with epidemiology and statistics which involve a lot of mathematical application.

Amidst the figures of risks, benefits, odds, ratios and so on, it's very common for people to be misled by default or by design. Take this example:
They [Tom Fahey and colleagues] wrote to 182 board members of district health authorities in England (all of whom would be in some way responsible for making important health service decisions), asking them which of four different rehabilitation programmes for heart attack victims they would prefer to fund:
• Programme A reduced the rate of deaths by 20%;
• Programme B produced an absolute reduction in deaths of 3%;
• Programme C increased patients' survival rate from 84% to 87%;
• Programme D meant that 31 people needed to enter the programme to avoid one death.
"How to Read a Paper" by Trisha Greenhalgh, 1997
Before you read on, give it a second of thought - which one would you choose? Which one is the most effective, and which one is the least effective?

The book author, Greenhalgh, proceeded to tell us the right answer: all of them are describing the same effect. Many of the health authorities (and I believe many of us as well) didn't notice it:
Of the 140 board members who responded, only three spotted that all four "programmes" in fact related to the same set of results. The other 137 preferred one or other of the programmes, thus revealing (as well as their own ignorance) the need for better basic training in epidemiology for health authority board members.
As a numerical example, let's say when the patients do not enter any program, 84 out of 100 patients survived; while for the rehabilitation program, 87 out of 100 patients survived. Rate of death is reduced from 16% to 13%, and therefore:
• A) It's a (16-13/16) X 100% ≈ 20% death reduction .
• B) The absolute death reduction is 16% - 13% = 3%.
• C) From the figure directly.
• D) Since the absolute death reduction is 3%, it means that 3 out of 100 patients have benefited through the rehabilitation program. That ratio is equivalent to 1 in 30 patients saved (rounded to the nearest ten, considering the significant digits).
It's amazing just how manipulable our mind is when it comes to numbers. The next time when you see a promotion claiming "buy two free one", remember that it's equivalent to 33% discount. For some reasons, "buy two free one" might seem more attractive to certain people than, say, 40% discount.

Errm.

bluez_aspic said...

"It's amazing just how manipulable our mind is when it comes to numbers. The next time when you see a promotion claiming "buy two free one", remember that it's equivalent to 33% discount."

Still not quite the same :P

youngyew said...

Yup I knew that... :) If you need to buy three or more anyway, it would make no difference; but if you actually need only one, you would have wasted money buying stuff for the future (without considering inflation etc).

day-dreamer said...

Erm, not wanting to venture into the deeper part of your post, let me just give my two cents about the sugar thingy.

It all depends. If you are a heavy user of sugar and your stock will deplete fast enough, it would of course be a better choice to buy the 5kg pack. Otherwise, 1kg will just do.

Hehe.

koln_auhc said...

I haven't read much of this post but the moment you mentioned risks it got me thinking.

Let's say the doctor assures you that a procedure has a 1% failure rate. Does that pacify you at all?

What if the 99 people in front of you came out safe from the operation. Are you, therefore, statistically screwed?

Haha.

http://en.wikipedia.org/wiki/Fooled_by_Randomness

sophisticatedsoul said...

Gosh. Epidemiology. It drives me crazy as I don't really like too much stats. Anyway, it is interesting getting to know how to read a paper and marvel at the fact that epidemiologists actually go through so much to study a case - which is something I definitely lack patience and interest to do.

Voon Seng said...

i'm taking a class on behavioral economics, and it's really interesting as we do talk a lot about how framing/packaging works.

Say, there are 2 models of laptop, say, one is \$500, and another \$1000. If the company wants to sell more of the \$1000 model, all they need is just to bring in another more expensive model. (a lot of real life examples). Like McDs in Europe, introduce a single cheese burger for \$1, double for \$2. By introducing a triple cheese burger for \$3, the sales of the double cheeseburger went up.

And also, people's understanding of probability is not that good, and there's a common misconception of overestimating low prob and underestimating high prob...

youngyew said...

day-dreamer: Agreed! :)

koln_auhc: Haha yeah it's kinda related to the gambler's mentality too.

Say you have this "odd / even" choice in Roulette. You stood there and watched even came out for the last three rolls. So now you're about to bet - would you choose odd or even? Some would say that "hey see, there are so many consecutive evens, so it's likely to be even again"; but some would argue "hey all the even's chances have already been 'consumed', so now it should be odd already".

But in reality, the next roll is still going to be 50-50, if we assume that there are equally odd and even squares in the roulette (which isn't the case in real life).

So as for the 1% failure rate thingy, it really depends on the necessity of the operation and what it means to "fail". If it's a non-essential operation and failure means death, then I would think twice; but if it's a very major procedure without which I would live in agony for my whole life, while failure means the problem not getting fixed, then I would say "why not?"

sophisticatedsoul: Haha sharp eyes! Yeah stats can get confusing, but trust me, the more you understand them, the more you would like them. :P

And yeah, having done research for two months, I fully empathize with the "effort" point. :)

Voon Seng: Interesting! But I guess there's a threshold to how many expensive models you could bring in. Using your McD example, does it mean that if I bring in another quadruple cheese burger, I would rake in more money?