Maths Teaser

[mossoi]

Here's a little maths exercise that I found quite interesting.

 

Assume there is a test for a fatal illness that is proven to be 98% accurate (a wrong diagnosis is given only 1 in 50 times) and the innacuracy is equal for both false negatives and false positives. Also assume that 0.5% of the tested population have the illness (1 in 200). If you are tested and your doctor tells you that you have contracted the illness what is the likelyhood that this diagnosis is correct? Is it worth demanding a second opinion?

 

I'll post the answer and proof after we've had some responses.

[Kedas]

If it's a fatal illness I would get a second opinion anyway independant of the odds

[mossoi]

That is an acceptable answer for the second question. You still need to answer the first one.

[BrainMan]

err...

[Dave]

I hate these questions - mainly because I can't do them.

 

(If you can't tell, I dropped my probability B module).

 

Also assume that 0.5% of the tested population have the illness (1 in 200).

 

I think that's a pretty useless statistic, but I'm almost certainly wrong.

[Kedas]

-what is the likelyhood that this diagnosis is correct?

-Assume there is a test for a fatal illness that is proven to be 98% accurate

[Dave]

That's what I was thinking.

[Kedas]

OR

Not sick people

2% of 199 get wrong result 3.98

98% of 199 get the right resultl 195.02

 

Sick people

2% of 1 get the wrong result 0.02 --> 0.02/200*100 = 0.01%

98% of 1 get the right result 0.98

[bloodhound]

I think, if u descrive the events properly, and then bang in the numbers in the bayes therom, or theorem of total probability, u get the answer. but i am to lazy to do anything like that.

[mossoi]

OK, here's the proof and answer. Kedas, you were pretty close.

 

Let's take a sample of 10,000 people to test.

 

Out of these 10,000, 50 (0.5%) have the illness.

49 (98%) percent of these people will test positive.

1 (2%) will wrongly test negative.

 

This leaves 9,950 (99.5%) people who don't have the illness.

199 (2%) of them will test positive by error.

 

So we have a total of 248 (199 + 49) people who have tested positive for the illness.

The probability of any one of these people having the illness is 49/248 (the number of correct positive tests over the total). This is approximately 20%!

 

So there is hope for those that were diagnosed with the illness and a retest is certainly worthwhile.

 

 

This example should make one think twice about statistics that are thrown about everyday (mainly by the scaremongering popular media).

[Dave]

Yah, our stats lecturer did a similar thing for breast cancer screening; statistically, it's more or less pointless.

[wolfson]

probability you have the illness: 0.05

prob that it comes up accurate: 0.98

prob that it comes up not accurate: 0.02

therefore, prob it says you have the illness:-

0.05 x 0.98 = 0.049

 

probability you don't have the illness and its says of have the illness = 0.95 x 0.02 = 0.019

 

therefore, probability it comes up positive= 0.019+0.049 =0.068

 

and the probability that you actually have the illness, given that it comes up positive:-0.049/0.068 = 0.72

 

get a 2nd opinion? urm...I would.

[wolfson]

Oh sorry didnt realise the answer was posted sorry

[Daniel]

I have come across a problem like this, where the overall possibility of a false positive was 33 % or something.

 

Solve it using bayes rule I think.