Its Friday again so :Homework for an 18 ish year old.

[AnthonyGerrard]

Given that 1% of the population have a certain disease called fibronobiosis

A test has been developed which has the following results.

1)If you have fibronobiosis, the test has a 99% chance of returning the result "positive".
2)If you do not have fibronobiosis, the test has 2% chance of (falsely) giving the result "positive".

A person randomly chosen is subjected to the test. They get the result "positive".

But what is the probability they actually have fibronobiosis?

 

[Groundrush]

99% because it already tells you that

1)If you have fibronobiosis, the test has a 99% chance of returning the result "positive".

 

[AnthonyGerrard]

99% because it already tells you that

The question asks it the other way round - given you have a positive result - whats the chances the random chosen person has fibronobiosis. - ie it could be a false positive.

 

[Groundrush]

The question asks it the other way round - given you have a positive result - whats the chances the random chosen person has fibronobiosis. - ie it could be a false positive.

I did think that was too easy

 

[rodmc]

2%

maybe its just me but the wording looks like its meant to have you doing a lot of calculating when the answer is already there

 

[AnthonyGerrard]

2%

maybe its just me but the wording looks like its meant to have you doing a lot of calculating when the answer is already there

Thats not right. The answer is maybe counter intuitive. But the calculation once onto it is fairly basic maths.

 

[Brianwarnock]

Odds of 99-2

Brian

 

[Vassago]

It's more complicated than that.

I'm going to go with

99% do not have the disease, and of that 98% will test negative, while 2% will test positive.

1% do have the disease, and of that, 99% will test positive, while 1% will test negative.

.01 * .99 = 0.0099 + .99 * .02 = 0.0198 ===== 0.0297 probability of getting a positive. Divide that 0.0099 by 0.0297, there is a 33% chance that he has the disease.

I forgot how to math. Am I right?

 

[stopher]

33.3% ish. Damn Vass just beat me

 

[stopher]

33.3% ish. Damn Vass just beat me

Here's my explanation (which I think ?!? is the same as Vassagos) ...
We want to know...

P1: Probability of +ve and diseased
----------------------------------------------------------------
P2: Prob of +ve and not diseased + P1: Prob of +ve and diseased

P1 = 0.99 x 0.01
P2 = 0.99 x 0.02
soln = 0.01/ (0.02+0.01) having cancelled out the 0.99

There's a a theorem for this but I can't remember it. Bayes? Last time I had to think about it I was 18ish

 

[Vassago]

I didn't think about the 99% cancel out. lol, that would have made it much easier.

 

[stopher]

I didn't think about the 99% cancel out. lol, that would have made it much easier.

I didn't realise either. When I first did the calculation I did it on a calculator and was surprised it was 33.33r or 1/3. So I did the algebra.

Is this your website Vass : http://www.vassarstats.net/bayes.html

Don't know if the answer is right though.

Nice problem Anthony.

 

[Vassago]

I didn't use a site. I went with memory, although it hurt a little. lol

 

[AnthonyGerrard]

well done!

Some fascinating(to me anyway) real world examples here.

Very simple, but very easy to get confused at the same time!

http://en.wikipedia.org/wiki/Simpson's_paradox

 

[RainLover]

Given that 1% of the population have a certain disease called fibronobiosis

This is the first and only fact we are given about the % of people that have the disease.

A test has been developed which has the following results.
1)If you have fibronobiosis, the test has a 99% chance of returning the result "positive".
2)If you do not have fibronobiosis, the test has 2% chance of (falsely) giving the result "positive".
A person randomly chosen is subjected to the test. They get the result "positive".

Information about a faulty test which has no bearing on the facts given.

But what is the probability they actually have fibronobiosis?

The actual question asked.
The only indisputable fact we are given is in the opening line.

The answer can only be 1 in 100.

If we create different tests we may get different results from those tests. However if they do not return 1 in 100 then the test is flawed as it is in the original question.

 

[stopher]

Information about a faulty test which has no bearing on the facts given.

The question is asking.... if a person tests positive, what is the chance that he/she has the disease?

So you do have to consider incorrect test results i.e. those people who tested positive but don't have the disease. Take a look here.

Think of it this way. From the question we can determine that around 3 in 100 people will test positive. 2 of those 3 will be false positive and 1 of the 3 will be a true positive - hence the 33.3%.

What's interesting about fingerprint technology (the example in the link) is that you can change the sensitivity e.g. you can increase the sensitivity so that it is less likely for an incorrect match - handy if the security is for a bank. But in doing so, this increases the chance of a correct match being rejected - which can be a bit annoying if the technology is being used a your workplace clock-in and you end up queueing each day while genuine people try to get a positive match. The theorem can be used to decide the most appropriate level of sensitivity for the required security level.

Chris

 

[RainLover]

Chris

The question asked is "But what is the probability they actually have fibronobiosis?"

You have changed the question to "if a person tests positive, what is the chance that he/she has the disease?" You could again change the question again if you wanted to. This does not in any way change the actual question asked.

As I said we are given the answer in the opening statement. "Given that 1% of the population have a certain disease called fibronobiosis"

We could introduce several other tests that return different results. However there is one constant and that is "1% of the population have the disease."

This is like throwing a dice. You have a 1 in 6 chance that a certain number will show on top. Now we could create some tests. In fact we could create hundreds of tests. Some correct some close and others way off.

The tests do not change the fact that "You have a 1 in 6 chance that a certain number will show on top"

 

[stopher]

Chris

The question asked is "But what is the probability they actually have fibronobiosis?"

No. The question is "A person randomly chosen is subjected to the test. They get the result "positive".

But what is the probability they actually have fibronobiosis?"

You are only reading half the question. "they" is referring only to a person who has tested positive, not the entire population.

You have changed the question to "if a person tests positive, what is the chance that he/she has the disease?" You could again change the question again if you wanted to. This does not in any way change the actual question asked.

No. I just re-worded it so that it was easier to understand.

 

[AnthonyGerrard]

Chris

The question asked is "But what is the probability they actually have fibronobiosis?"

You have changed the question to "if a person tests positive, what is the chance that he/she has the disease?" You could again change the question again if you wanted to. This does not in any way change the actual question asked.

As I said we are given the answer in the opening statement. "Given that 1% of the population have a certain disease called fibronobiosis"

We could introduce several other tests that return different results. However there is one constant and that is "1% of the population have the disease."

This is like throwing a dice. You have a 1 in 6 chance that a certain number will show on top. Now we could create some tests. In fact we could create hundreds of tests. Some correct some close and others way off.

The tests do not change the fact that "You have a 1 in 6 chance that a certain number will show on top"

Theres a test that 100% accuate that the result is odd or even.

The result of the test is that the number thrown tests positive to being odd.

Whats the chance given that the test result is +ve to being odd, that the number thrown is a 3?

Its not 1/6 is it.

 

[RainLover]

The following is a copy of the original.

A person randomly chosen is subjected to the test. They get the result "positive".

But what is the probability they actually have fibronobiosis?

If these two lines were meant to be one question I would have thought that they were contained in the same paragraph. However it does not matter one little bit.

We are given the answer to the question and that is 1 in 100. This is regardless of what this test or any other test returns.

How you change "1 in 100" to "1 in 3" is beyond belief.