Thursday, 12 September 2013

Goats and a Cadillac

The BBC runs, for the umpteenth time, a story about the Monty Hall Problem, with which I am sure you are all familiar.

It ends with this:

Next year we will need something different, perhaps Simpson's Paradox. Imagine that 1% of people have a certain disease.

A diagnostic test has been developed which performs as follows - if you have the disease, the test has a 99% chance of giving the result "positive", while if you do not have the disease, the test has 2% chance of (falsely) giving the result "positive".

A randomly chosen person takes the test. If they get the result "positive", what is the probability that they actually have the disease? The answer, 1/3, is perhaps surprisingly low.

Answers on a postcard please.


Simple.

If you take a person at random, there is a 99% chance he does not have the disease, but will test will show a positive in 2% of cases anyway.

There is a 1% chance he does have the disease, and the test will be correct 99% of the time.

So if we test 100 people at random, there will be 1.98* false positives and 0.99** true positives, so two-thirds of those 2.97 positives are false.

* 100 people x 99% disease-free x 2% false positives = 1.98
** 100 people x 1 with the disease x 99% true positives = 0.99

4 comments:

Lola said...

Look. I'm knackered. I am having to negotiate with Aviva. I've got staff nagging me. And then I get brain pain from puzzles like this...

Dinero said...

That description of the monty hall problem on the BBC site was a bit confused.

People don't switch because they think the probabilaty is 50/50, the reason they don't switch is because they think it is 1/3 throughout. If the noticed they were being given a 50/50 choice at one point then they would swith as that is an improvement over 1/3



Mark Wadsworth said...

L, doing maths puzzles is how I relax.

D, their explanation is as good as any I've seen:

"If you stick with your first choice, you will end up with the Caddy if and only if you initially picked the door concealing the car.

If you switch, you will win that beautiful automobile if and only if you initially picked one of the two doors with goats behind them.

If you can accept this logic then you're home and dry, because working out the odds is now as easy as pie - sticking succeeds 1/3 of the time, while switching works 2/3 of the time."

Lola said...

MW. I do cryptic crosswords as part of my relaxation, but I also get out a lot...