Probability and Statistical Reversal Paradoxes
Created | Updated Nov 14, 2008
Humans are rubbish at understanding probabilities and statistics. We rely a lot of the time on gut feelings to make decisions, but our guts are often wrong about situations involving probability and statistics. Casinos, bookmakers, sideshow operators and the National Lottery depend on this perceptual blind spot for their income.
Imagine someone tosses a coin in the air, and it comes up heads. Not surprising in itself.
Imagine they do it again, and it comes up heads again. Again, fine.
Imagine they do it seven more times, and every time it comes up heads. By now, if you've any money riding on the outcome, you're surely feeling a bit itchy. Surely a tail must come up soon.
The tosser1 offers you a bet. If it comes up heads, you give him ten pounds. If it comes up tails, he'll give you eight pounds. You ask him why not ten. He replies that it's surely time for a tail by now, so you must have a better than even chance. He tosses the coin...
After a long run of heads, what are the odds of a tail? There are two possible answers to this:
They're exactly the same as the odds were for the first throw - 50/50. The bet is not fair because the odds do not depend on the outcome of previous throws, as your 'friend' is trying to suggest.
They're exactly the same as the odds were for the first throw - zero, because your friend is tossing a double headed coin. The bet is not fair, but your friend has to pay for that gimmicked coin somehow.
Jelly Beans and Statistical Reversal
Dave likes jelly beans. He likes lemon ones and blackcurrant ones, so they're the only flavours he ever buys. He prefers lemony ones, but he's not that fussy, really.
He sits in front of his computer with two piles of jelly beans on the desk. On the left, five lemon and six blackcurrant. On the right, three lemon and four blackcurrant. Dave wants a jelly bean, but he is engrossed in an h2g2 Entry about paradoxes, and can't tear his eyes away from the screen. Given his preference for lemon jelly beans, which pile should he grope for?
Obviously, in this case, the pile on the left. He has a 45% chance of getting a lemon jelly bean if he gropes off to the left, and only a 43% chance if he reaches right. So far, so easy.
There are two other piles on the desk. On the far left, a pile with six lemon and three blackcurrant. On the far right, nine lemon and five blackcurrant. Still engrossed, Dave wants a jelly bean from one of these two piles. Groping left offers a 67% chance of a lemon jelly bean, while reaching right offers only a 64% chance.
So far this is simple enough. But disaster strikes when Dave's anally-retentive girlfriend bustles into the room and 'tidies up'. She sweeps all the jelly beans on the left into a single pile, and all the ones on the right into a single pile.
So now our hero wants a jelly bean, and would still prefer a lemon one. In either of the two cases above, reaching left would give him the best chance of getting what he wants, so obviously he's going to want to reach left, yes?
Wrong. Reaching left now offers a 55% chance of a lemon bean, while reaching right gives 57%. By the simple act of tidying up, Dave's girlfriend has managed to reverse the probabilities and defy common sense!
Statistical reversal is an important phenomenon with applications in many fields far more important than jelly beans.
Imagine a doctor who wants to test the efficacy of a new treatment. He has patients in London and Birmingham on whom he wishes to run a clinical trial. He consults a statistician, who advises him to treat 91% of the London patients with the new drug, and the other 9% with the old one, selecting randomly who is to get what. Similarly, he advises giving the new drug to one patient in Birmingham for every 100 patients who are treated with the old drug.
The doctor goes away, tries the drug out and summarises the results.
|Effectiveness of treatment||London||London||Birmingham||Birmingham|
|Old drug||New Drug||Old Drug||New Drug|
|Not effective||950 (95%)||9000 (90%)||5000 (50%)||5 (5%)|
|Effective||50 (5%)||1000 (10%)||5000 (50%)||95 (95%)|
He's quite happy - the new drug seems to him almost twice as effective as the old one. He's all ready to write a positive report to the drug company, when he gets a message from the statistician, telling him to stop prescribing the drug right away. His table of results (using exactly the same raw data) looks like this:
|Effectiveness||Old drug||New drug|
|Not effective||5,950 (54%)||9,005 (89%)|
|Effective||5,050 (46%)||1,095 (11%)|
This is known as 'Simpson's Paradox', after E. H. Simpson, who first wrote about it in 1951.
Who is right? In this case, the doctor. In the example, there were 6,145 recoveries. If the new treatment had been universal, we could expect 10,700. If only the old treatment were used, we could expect 5,600. The new treatment really is significantly better.
So why does the statistician see what he does? Because he has failed to take into account that the treatment was given to a much larger proportion of the London patients, and they are statistically less likely to recover regardless of their treatment. This skews the combined results.
As the introduction stated, people are very poor at judging probabilities. The introduction of statistics just makes it harder to judge - common sense ceases to be a useful guide even with fairly simple examples like the piles of jelly beans. The possibility that we are not considering all the possible contributory factors makes life even more difficult. Basically, in any situation involving probabilities and statistics - never rely solely on your common sense and intuition. Do the maths.