Test ur logic, Probability question

pif · #1 08-25-2005, 06:32 AM

*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]

PairTheBoard · #2 08-25-2005, 06:57 AM

[ 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

xniNja · #3 08-26-2005, 07:02 PM

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%.

BruceZ · #4 08-26-2005, 07:31 PM

[ 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.

xniNja · #5 08-26-2005, 07:39 PM

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?

BruceZ · #6 08-26-2005, 08:28 PM

[ 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?

[/ QUOTE ]

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.

xniNja · #7 08-26-2005, 08:39 PM

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

[/ QUOTE ]

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.

BruceZ · #8 08-26-2005, 08:52 PM

[ 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:

[ 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%.

[/ QUOTE ]

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.

xniNja · #9 08-26-2005, 09:02 PM

[ 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?

BruceZ · #10 08-26-2005, 09:10 PM

[ 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?

[/ QUOTE ]

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?