Test ur logic, Probability question
 

*Given that in some population a 1% of them r infected by some disease.
*And given that there is a medical test that can recognize the disease, the test results r positive or negative, But the test results r true only in 90% from the tests.

*The Problem: a man from this population made the test, and the result was Positive(sick), what the odds that this man is Realy sick? [img]/images/graemlins/confused.gif[/img]

Re: Test ur logic, Probability question
 

[ QUOTE ]
*Given that in some population a 1% of them r infected by some disease.
*And given that there is a medical test that can recognize the disease, the test results r positive or negative, But the test results r true only in 90% from the tests.

*The Problem: a man from this population made the test, and the result was Positive(sick), what the odds that this man is Realy sick? [img]/images/graemlins/confused.gif[/img]

[/ QUOTE ]

.009/(.009+.099)

PairTheBoard

Re: Test ur logic, Probability question
 

I think it's just 90%. The 1% of the population being sick doesn't tell you anything.

90% of the time the test is true, therefore the one time it was tested positive, it should be accurate to a degree of 90%.

Re: Test ur logic, Probability question
 

[ QUOTE ]
I think it's just 90%. The 1% of the population being sick doesn't tell you anything.

90% of the time the test is true, therefore the one time it was tested positive, it should be accurate to a degree of 90%.

[/ QUOTE ]

PairTheBoard is correct. The odds are 11:1 against the man being sick.

Here's an easy way to do these Bayes' theorem problems without messing with formulas. There are 2 possible scenarios. First, the man could be healthy (99% chance) and the test could be wrong (10% chance), and the chance of that scenario is 99% * 10% = 9.9%. Second, the man could really be sick (1% chance) and the test could be right (90% chance), and the chance of that scenario is 1% * 90% = 0.9%. So the odds are 9.9:0.9 or 11:1 against him being sick.

Re: Test ur logic, Probability question
 

Yea, but the post said test your logic - not application of Bayes Theoreom, although he does say probability question... I found it more interesting to attempt a logical (but not technically correct) solution, but you're right I failed since the post said "Odds" of him being sick, and not the test being right. It was fun nonetheless.

If he wanted a Bayes Theorem answer, why's this post here?

Re: Test ur logic, Probability question
 

[ QUOTE ]
Yea, but the post said test your logic - not application of Bayes Theoreom, although he does say probability question... I found it more interesting to attempt a logical (but not technically correct) solution, but you're right I failed since the post said "Odds" of him being sick, and not the test being right. It was fun nonetheless.

If he wanted a Bayes Theorem answer, why's this post here?

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This can be solved logically by persons who never heard of Bayes' theorem; however, if they don't come up with the same answer as Bayes' theorem, then their answer isn't logical; it's wrong.

The test is right 90% of the time. It doesn't have a 90% chance of being right when it pronounces someone sick. Once it pronounces someone sick, it only has a 1/12 or 8.3% chance of being right. When it pronounces someone well, it has almost a 99.9% chance of being right.

The fact that 1% of the people are sick is not irrelevant information; it's crucial for determining the odds that the test is right.

Re: Test ur logic, Probability question
 

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The test is right 90% of the time. It doesn't have a 90% chance of being right when it pronounces someone sick.

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You're obviously right mathematically, I'm not trying to say the number is wrong. This is a question of semantics... I was mereley arguing that the following series of words is also a correct and logical statement:

A test is right 90% of the time. Someone takes the test and receives a result. Therefore, 90% of the time, the result is right.

I make no attempt to disprove Bayes Theorem, but argue that syntax and semantics are valid components for logical theory.

Re: Test ur logic, Probability question
 

[ QUOTE ]
I was mereley arguing that the following series of words is also a correct and logical statement:

A test is right 90% of the time. Someone takes the test and receives a result. Therefore, 90% of the time, the result is right.

[/ QUOTE ]

Not if that result is that he is sick. That was the question. If you ignore the result, then you are not taking into account all of the relevant information.

You said:

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90% of the time the test is true, therefore the one time it was tested positive, it should be accurate to a degree of 90%.

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This is false. A correct statement is that if 90% of the time the test is true, then when it tests positive it has an 8.3% chance of being accurate. To conclude this, you need to use the fact that 1% of the population is really sick. But you said:

[ QUOTE ]
The 1% of the population being sick doesn't tell you anything.

[/ QUOTE ]

This is false, and it led you to the wrong conclusion.

Re: Test ur logic, Probability question
 

[ QUOTE ]

90% of the time the test is true, therefore the one time it was tested positive, it should be accurate to a degree of 90%.

This is false. A correct statement is that if 90% of the time the test is true, then when it tests positive it has an 8.3% chance of being accurate.


[/ QUOTE ]

You're missing my point. The statement is true by itself. Now I admit I didn't answer the original question (what are the odds of that particular case result being correct, given the scenario), but it is still a valid, logical, and true statement.

Perhaps more clearly stated: If the probability of a test being correct is 90%, then the probability of a test being correct is 90%, which means that on any given test the probability of the test being correct is 90%.

This holds true logically- as a construct, not as a mathematical equation with 1% of the population being sick. Follow?

Re: Test ur logic, Probability question
 

[ QUOTE ]
[ QUOTE ]

90% of the time the test is true, therefore the one time it was tested positive, it should be accurate to a degree of 90%.

This is false. A correct statement is that if 90% of the time the test is true, then when it tests positive it has an 8.3% chance of being accurate.


[/ QUOTE ]

You're missing my point. The statement is true by itself. Now I admit I didn't answer the original question (what are the odds of that particular case result being correct, given the scenario), but it is still a valid, logical, and true statement.

Perhaps more clearly stated: If the probability of a test being correct is 90%, then the probability of a test being correct is 90%, which means that on any given test the probability of the test being correct is 90%.

This holds true logically- as a construct, not as a mathematical equation with 1% of the population being sick. Follow?

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I follow that you have stated a useless logical tautology which has nothing to do with the problem at hand. Go back and read what you just wrote which I have bolded. Is this really what you intend to say?

Re: Test ur logic, Probability question
 

Yes, I repeated it for emphasis. I don't see why it is useless or a tautology. Anyone can plug in 2 numbers into Baye's Theoreom... You claim I'm ignoring the relevant information of 1% of the pop. being sick, but at the same time I think it's an equally valid claim, and if not correct, interesting and logical approach to claim the reverse. It's a conceivable logical argument that if the results of a test are 90% accurate, then any result is 90% accurate- I don't see this as tautology.

Re: Test ur logic, Probability question
 

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Yes, I repeated it for emphasis. I don't see why it is useless or a tautology.

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Oh come on. You don't see why it's useless to state that If the probability of a test being correct is 90%, then the probability of a test being correct is 90%? You've just asserted the reflexive property that A = A. Congratulations.

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Anyone can plug in 2 numbers into Baye's Theoreom

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Not so. A very small percentage of people can plug in 2 numbers (actually 4 numbers in this case) into Bayes' theorem correctly. Most people can't even spell "Baye's Theoreom".


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You claim I'm ignoring the relevant information of 1% of the pop. being sick, but at the same time I think it's an equally valid claim, and if not correct, interesting and logical approach to claim the reverse. It's a conceivable logical argument that if the results of a test are 90% accurate, then any result is 90% accurate- I don't see this as tautology.

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No *that's* not a tautology. That's a false statement. Any result is not 90% accurate. The only results that are 90% accurate are the results which are unknown. There were no unknown results in this problem. There was but one result in this problem, and it was clearly stated that the result was positive. Yet you stated in your original post that this *positive* result was 90% accurate. WRONG.

If you want to ignore the conditions of a problem, make false statements, and amuse yourself by stating the reflexive property, fine, go ahead. But don't do it in response to a clearly defined problem, and then try to state that what you have done is in some way relevant. I have no more time for this.

Re: Test ur logic, Probability question
 

I immediately retracted my first post... but fine, you're right... It's true I was much more concerned with my amusement of an antithetical construct than actually solving the problem, but that post was definitely not clearly defined, and I wasn't merely quoting the reflexive property.

Re: Test ur logic, Probability question
 

I think it is 90%.

If this question was a Bayes Theorem Question, it was very poorly worded.

What does, "the test results are true only 90% of the time" mean?

Is that the rate of false positives to false negatives? What does it mean?

Does it mean if you select a random test result, it's true 90% of hte time?

Re: Test ur logic, Probability question
 

[ QUOTE ]
I think it is 90%.

If this question was a Bayes Theorem Question, it was very poorly worded.

What does, "the test results are true only 90% of the time" mean?

Is that the rate of false positives to false negatives? What does it mean?

Does it mean if you select a random test result, it's true 90% of hte time?

[/ QUOTE ]

I took it to mean that it will call a well person well 90% of the time, and it will call a sick person sick 90% of the time. These two probabilities should have been specified separately, but since they weren't, the obvious assumption is that they are both 90%. If it right in 90% of well people, then it must be right in 90% of sick people too if it is to be right in 90% of all people. Then this is clearly a Bayes' theorem problem with an answer of 11:1 against being sick.

A real world problem might have 10% false positives, and no false negatives, and it is possible that the OP miscopied this problem. In this case, the chance that the man is really sick would be 1%*100% = 1%, and the chance that he is well would be 99% * 10% = 9.9%. Then the odds against him being sick would be 9.9:1.

Re: Test ur logic, Probability question
 

See, I wasn't the only one. The poor English and "u r is tru" and such is deceptive! Does this also prove the 90% 1st-priority logic is reasonable? Maybe not sound, but it's logical.

If it is a standard Bayes' Theorem problem, that's fine- you're correct. Learn to be able to see other viewpoints on the same problem.

Re: Test ur logic, Probability question
 

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pif --
But the test results r true only in 90% from the tests

[/ QUOTE ]

My opinion now is that this problem is technically not well defined because this phrase can technically be translated in two ways. The phrase "from the tests" seems redundant and unnecessary or else impossible to glean precise meaning from. Dropping that phrase the statement becomes,

'But the test results r true only in 90% '

The words "but" and "only" just serve to emphasize that the test is not 100% accurate, so they can be dropped. We already know that.

'the test results r true in 90%'

Now, what does this mean? Does it mean that whatever the test result Is there is a 90% chance that it is True? In other words, If the test result says you are sick there is a 90% chance you are sick, and if the test result says you are ok there is a 90% chance that you are ok? Technically I don't think you can rule that out. But if that's what pif means then why does he ask the chances of being sick when the test result says you are sick? There's no point to it. He's just wasting our time if that's what he means. Although you might get a real problem out of it by asking what the False Positive and False Negative probabilties would have to be for the test to yield such a 90% overall accuracy in this population.

So, assuming pif does not intend to waste our time, what can we Infer the vauge statement means. As Bruce and I have done, the natural inference given the context is that when pif says essentially,

'the test results r true in 90%'

he means the Test gives 10% False positives and 10% False Negatives.

A good example for how difficult it is to get a computer to understand people language.

PairTheBoard