If it says it's positive, the chances of you having the disease is 99% because the test is 99% accurate: if it says it's positive, it's right 99 times out of 100.
The question is badly worded. I suspect it intends to ask, what's the chance of having the disease and the test showing it up? Quite different. In that case it's 1% (chance of having the disease) x 99% chance of the test showing it up, which is 0.99%.
I see your point, but the question is intended to (and I think does) ask, "If you get a positive result, what is the chance that you actually have the disease?".
99% of the population won't have the disease, but 1% of that 99% will display a false positive, so 0.99% of the whole population will have a positive result but not actually have the disease.
Also, out of the 1% that have the disease, 99% will show as a positive, so 0.99% of the population will have a positive result and have the disease.
So it is 50/50, as 98.01% of the population will correctly show a negative result and 0.01% of the population will incorrectly show a negative result. That all adds up to 100%.
I think I have that right!
This is one of those puzzling puzzles where logic appears to defeat intuition. However, the solution to the conundrum is often in the order of events and I think that's what's going on here.
I think what you are saying is that before I take the test I have a 0.99% chance of a false positive, i.e. I don't have the disease, but I have a positive test result. Also, that I have the same probability, i.e. 0.99%, of having the disease and getting a positive test result. When we add the ex ante odds of a correct negative (98.01%) and a false negative (0.01%) we get 100%.
That's fine, but these odds change completely once I've taken the test. There are now only two possible outcomes. The test is right or it's wrong. Unfortunately, for anyone who tests positive, the test is right nearly all of the time. Out of every hundred people who have tested positive, only one will turn out not to have the disease.
99% of the population won't have the disease, but 1% of that 99% will display a false positive, so 0.99% of the whole population will have a positive result but not actually have the disease.
Also, out of the 1% that have the disease, 99% will show as a positive, so 0.99% of the population will have a positive result and have the disease.
So it is 50/50, as 98.01% of the population will correctly show a negative result and 0.01% of the population will incorrectly show a negative result. That all adds up to 100%.
I think I have that right!
This is one of those puzzling puzzles where logic appears to defeat intuition. However, the solution to the conundrum is often in the order of events and I think that's what's going on here.
I think what you are saying is that before I take the test I have a 0.99% chance of a false positive, i.e. I don't have the disease, but I have a positive test result. Also, that I have the same probability, i.e. 0.99%, of having the disease and getting a positive test result. When we add the ex ante odds of a correct negative (98.01%) and a false negative (0.01%) we get 100%.
That's fine, but these odds change completely once I've taken the test. There are now only two possible outcomes. The test is right or it's wrong. Unfortunately, for anyone who tests positive, the test is right nearly all of the time. Out of every hundred people who have tested positive, only one will turn out not to have the disease.
However, due to the fact that we know that only 1% of the population actually has the disease and the vast majority do not, there are going to be far more false positives than there are going to be false negatives. Hence why the test is so accurate...most people do not have the disease and in most cases the correct result is given. Likewise if you actually have the disease, then there is a 99% chance of you being correctly identified.
Out of 10,000 people, 9,801 will correctly test negative, 99 will be a false positive, 99 will be a true positive and 1 will be a false negative.
Yes, you are correct, the odds of you actually having the disease when you test positive do change completely...from 1% to 50%.
The error you make is assuming that if you test positive, you have a 99% chance of having the disease. That's not what it means when the test is 99% accurate. The test is 99% accurate because 99% of those who don't have the disease will test negative and 99% of those who have the disease will test positive.
99% of the population won't have the disease, but 1% of that 99% will display a false positive, so 0.99% of the whole population will have a positive result but not actually have the disease.
Also, out of the 1% that have the disease, 99% will show as a positive, so 0.99% of the population will have a positive result and have the disease.
So it is 50/50, as 98.01% of the population will correctly show a negative result and 0.01% of the population will incorrectly show a negative result. That all adds up to 100%.
I think I have that right!
This is one of those puzzling puzzles where logic appears to defeat intuition. However, the solution to the conundrum is often in the order of events and I think that's what's going on here.
I think what you are saying is that before I take the test I have a 0.99% chance of a false positive, i.e. I don't have the disease, but I have a positive test result. Also, that I have the same probability, i.e. 0.99%, of having the disease and getting a positive test result. When we add the ex ante odds of a correct negative (98.01%) and a false negative (0.01%) we get 100%.
That's fine, but these odds change completely once I've taken the test. There are now only two possible outcomes. The test is right or it's wrong. Unfortunately, for anyone who tests positive, the test is right nearly all of the time. Out of every hundred people who have tested positive, only one will turn out not to have the disease.
However, due to the fact that we know that only 1% of the population actually has the disease and the vast majority do not, there are going to be far more false positives than there are going to be false negatives. Hence why the test is so accurate...most people do not have the disease and in most cases the correct result is given. Likewise if you actually have the disease, then there is a 99% chance of you being correctly identified.
Out of 10,000 people, 9,801 will correctly test negative, 99 will be a false positive, 99 will be a true positive and 1 will be a false negative.
Yes, you are correct, the odds of you actually having the disease when you test positive do change completely...from 1% to 50%.
The error you make is assuming that if you test positive, you have a 99% chance of having the disease. That's not what it means when the test is 99% accurate. The test is 99% accurate because 99% of those who don't have the disease will test negative and 99% of those who have the disease will test positive.
You're getting a bit confused. There may be 2 outcomes, but that's also true if you've got 99 red balls in a bag and 1 green one - you're still more probable to get a red one. The probability of getting a red ball is 99%.
So in 100 tests, 99 are correct and 1 is wrong - regardless of positive/negative, says the question. There are indeed two outcomes, but 99 times out of 100 you're getting the right answer.
I see what you're trying to say, but there's 100% probability that you're incorrect.
The question doesn't indicate whether the quoted test accuracy already factors in the fact that we know that only 1% of the population have the disease.
If you think about it the test is actually pretty useless! You might just as well take a look at each person and say "You don't have the disease". This will be correct 99% of the time and have the same accuracy as if you bother to actually do the test.
So my guess would be 99%. Why would the devisors of the test quote a 99% accuracy figure for a positive test if they knew that that meant you only had a 50% chance of having the disease?
99% of the population won't have the disease, but 1% of that 99% will display a false positive, so 0.99% of the whole population will have a positive result but not actually have the disease.
Also, out of the 1% that have the disease, 99% will show as a positive, so 0.99% of the population will have a positive result and have the disease.
So it is 50/50, as 98.01% of the population will correctly show a negative result and 0.01% of the population will incorrectly show a negative result. That all adds up to 100%.
I think I have that right!
This is one of those puzzling puzzles where logic appears to defeat intuition. However, the solution to the conundrum is often in the order of events and I think that's what's going on here.
I think what you are saying is that before I take the test I have a 0.99% chance of a false positive, i.e. I don't have the disease, but I have a positive test result. Also, that I have the same probability, i.e. 0.99%, of having the disease and getting a positive test result. When we add the ex ante odds of a correct negative (98.01%) and a false negative (0.01%) we get 100%.
That's fine, but these odds change completely once I've taken the test. There are now only two possible outcomes. The test is right or it's wrong. Unfortunately, for anyone who tests positive, the test is right nearly all of the time. Out of every hundred people who have tested positive, only one will turn out not to have the disease.
However, due to the fact that we know that only 1% of the population actually has the disease and the vast majority do not, there are going to be far more false positives than there are going to be false negatives. Hence why the test is so accurate...most people do not have the disease and in most cases the correct result is given. Likewise if you actually have the disease, then there is a 99% chance of you being correctly identified.
Out of 10,000 people, 9,801 will correctly test negative, 99 will be a false positive, 99 will be a true positive and 1 will be a false negative.
Yes, you are correct, the odds of you actually having the disease when you test positive do change completely...from 1% to 50%.
The error you make is assuming that if you test positive, you have a 99% chance of having the disease. That's not what it means when the test is 99% accurate. The test is 99% accurate because 99% of those who don't have the disease will test negative and 99% of those who have the disease will test positive.
You're getting a bit confused. There may be 2 outcomes, but that's also true if you've got 99 red balls in a bag and 1 green one - you're still more probable to get a red one. The probability of getting a red ball is 99%.
So in 100 tests, 99 are correct and 1 is wrong - regardless of positive/negative, says the question. There are indeed two outcomes, but 99 times out of 100 you're getting the right answer.
I see what you're trying to say, but there's 100% probability that you're incorrect.
You're confusing the premise of the test - someone already has or has not gotten the disease. Your analogy would require 99 people to already have a red ball, 1 to have a green ball, and a psychic who is correct 99% of the time (we also need to make the assumption that the psychic has no idea how many red or green balls there are). Odds are that he will guess everyone correctly but if he guesses someone has the green ball, there is actually just as much likelihood that he is wrong as that person having the green ball (since both are 1% chances in their respective probabilities).
The question doesn't indicate whether the quoted test accuracy already factors in the fact that we know that only 1% of the population have the disease.
If you think about it the test is actually pretty useless! You might just as well take a look at each person and say "You don't have the disease". This will be correct 99% of the time and have the same accuracy as if you bother to actually do the test.
So my guess would be 99%. Why would the devisors of the test quote a 99% accuracy figure for a positive test if they knew that that meant you only had a 50% chance of having the disease?
The test has no idea what the incidence of disease is in the population though. Consider a blood test - a blood test has no idea how rare certain blood groups are because it is only testing based on the make-up of the blood that is being tested.
The common mistake everyone is making is that you are not factoring in the sheer number of false positives there will be in the non-diseased population.
The test is 99% accurate because if you actually have the disease, there is a 99% chance you will be correctly diagnosed. This does not take into account the actual incidence of disease in the population. The reason why it is 50-50 because you have no idea if you have the disease or not and a false positive is just as likely when you factor in both the accuracy and incidence as a correct positive.
99% of the population won't have the disease, but 1% of that 99% will display a false positive, so 0.99% of the whole population will have a positive result but not actually have the disease.
Also, out of the 1% that have the disease, 99% will show as a positive, so 0.99% of the population will have a positive result and have the disease.
So it is 50/50, as 98.01% of the population will correctly show a negative result and 0.01% of the population will incorrectly show a negative result. That all adds up to 100%.
I think I have that right!
This is one of those puzzling puzzles where logic appears to defeat intuition. However, the solution to the conundrum is often in the order of events and I think that's what's going on here.
I think what you are saying is that before I take the test I have a 0.99% chance of a false positive, i.e. I don't have the disease, but I have a positive test result. Also, that I have the same probability, i.e. 0.99%, of having the disease and getting a positive test result. When we add the ex ante odds of a correct negative (98.01%) and a false negative (0.01%) we get 100%.
That's fine, but these odds change completely once I've taken the test. There are now only two possible outcomes. The test is right or it's wrong. Unfortunately, for anyone who tests positive, the test is right nearly all of the time. Out of every hundred people who have tested positive, only one will turn out not to have the disease.
However, due to the fact that we know that only 1% of the population actually has the disease and the vast majority do not, there are going to be far more false positives than there are going to be false negatives. Hence why the test is so accurate...most people do not have the disease and in most cases the correct result is given. Likewise if you actually have the disease, then there is a 99% chance of you being correctly identified.
Out of 10,000 people, 9,801 will correctly test negative, 99 will be a false positive, 99 will be a true positive and 1 will be a false negative.
Yes, you are correct, the odds of you actually having the disease when you test positive do change completely...from 1% to 50%.
The error you make is assuming that if you test positive, you have a 99% chance of having the disease. That's not what it means when the test is 99% accurate. The test is 99% accurate because 99% of those who don't have the disease will test negative and 99% of those who have the disease will test positive.
You're getting a bit confused. There may be 2 outcomes, but that's also true if you've got 99 red balls in a bag and 1 green one - you're still more probable to get a red one. The probability of getting a red ball is 99%.
So in 100 tests, 99 are correct and 1 is wrong - regardless of positive/negative, says the question. There are indeed two outcomes, but 99 times out of 100 you're getting the right answer.
I see what you're trying to say, but there's 100% probability that you're incorrect.
You're confusing the premise of the test - someone already has or has not gotten the disease. Your analogy would require 99 people to already have a red ball, 1 to have a green ball, and a psychic who is correct 99% of the time (we also need to make the assumption that the psychic has no idea how many red or green balls there are). Odds are that he will guess everyone correctly but if he guesses someone has the green ball, there is actually just as much likelihood that he is wrong as that person having the green ball (since both are 1% chances in their respective probabilities).
So in fact I am 100% right
Bugger, you are as well. You're right, I misread the question to mean it was known he had the disease. It actually says that he had a positive result.
So in a sample of 10,000 only 100 people will have it, which means 99 people will have positive test results. But that also means 9900 don't have it, and the inaccuracy of the test means 1% of those - another 99 - will also get positive test results. So from the sample of 198 who had positive results, half of them will actually have it. 50%.
Damn! I studied maths/stats at uni, I hang my head in shame ;-)
EDIT: I also now realise it wasn't you who talked about outcomes, bloody quoting...
Damn! I studied maths/stats at uni, I hang my head in shame ;-)
Haha to be fair so did I but I didn't get the answer straight away. The puzzle tricks you into thinking that the test takes into account the incidence of disease, as well as making you think that a positive result means there is 99% of having the disease, rather than the other way round (having the disease means there is a 99% of a positive result).
I miss the puzzle thread we had a few months ago, definitely some good brain-teasers.
Damn! I studied maths/stats at uni, I hang my head in shame ;-)
Haha to be fair so did I but I didn't get the answer straight away. The puzzle tricks you into thinking that the test takes into account the incidence of disease, as well as making you think that a positive result means there is 99% of having the disease, rather than the other way round (having the disease means there is a 99% of a positive result).
I miss the puzzle thread we had a few months ago, definitely some good brain-teasers.
The maths takes it into account, but you're right - the test is not the event you're looking for. The chances of having any type of positive result is 1.98% (true 1% x 99% + false 99% x 1%), the chance of it being a true positive is 0.99%, hence 50% probability given you had a positive result. Counter intuitive, because it's easy to forget you need to divide the event you're looking for by all possibilities.
If it says it's positive, the chances of you having the disease is 99% because the test is 99% accurate: if it says it's positive, it's right 99 times out of 100.
The question is badly worded. I suspect it intends to ask, what's the chance of having the disease and the test showing it up? Quite different. In that case it's 1% (chance of having the disease) x 99% chance of the test showing it up, which is 0.99%.
The question is worded correctly and is not trying to ask your version.
If it says it's positive, the chances of you having the disease is 99% because the test is 99% accurate: if it says it's positive, it's right 99 times out of 100.
The question is badly worded. I suspect it intends to ask, what's the chance of having the disease and the test showing it up? Quite different. In that case it's 1% (chance of having the disease) x 99% chance of the test showing it up, which is 0.99%.
The question is worded correctly and is not trying to ask your version.
Yep, we've already got there. Thanks for your contribution ;-)
There are 7 pirates on a ship including the captain. They have an order of rank that they all know. So the captain is rank 1 and the least important is rank 7. At any time the pirates can take a vote to remove the Captain, under which circumstances the rank 2 pirate becomes captain. If the vote is tied then the current captain has the deciding vote.
They come across 100 coins (of equal value), the captain must decide how to distribute these coins among the crew. What is the most number of coins he can take for himself without a mutiny?
I should add that the pirates are all mathematical geniuses and completely ruthless with no loyalty whatsoever.
Fiiish obviously googled the working-out before I did though
Leuth I have been trying to find out the probability if the person took the test a second time and came out positive. Nobody on here has answered that. Can you enlighten us?
Comments
I think what you are saying is that before I take the test I have a 0.99% chance of a false positive, i.e. I don't have the disease, but I have a positive test result. Also, that I have the same probability, i.e. 0.99%, of having the disease and getting a positive test result. When we add the ex ante odds of a correct negative (98.01%) and a false negative (0.01%) we get 100%.
That's fine, but these odds change completely once I've taken the test. There are now only two possible outcomes. The test is right or it's wrong. Unfortunately, for anyone who tests positive, the test is right nearly all of the time. Out of every hundred people who have tested positive, only one will turn out not to have the disease.
Out of 10,000 people, 9,801 will correctly test negative, 99 will be a false positive, 99 will be a true positive and 1 will be a false negative.
Yes, you are correct, the odds of you actually having the disease when you test positive do change completely...from 1% to 50%.
The error you make is assuming that if you test positive, you have a 99% chance of having the disease. That's not what it means when the test is 99% accurate. The test is 99% accurate because 99% of those who don't have the disease will test negative and 99% of those who have the disease will test positive.
So in 100 tests, 99 are correct and 1 is wrong - regardless of positive/negative, says the question. There are indeed two outcomes, but 99 times out of 100 you're getting the right answer.
I see what you're trying to say, but there's 100% probability that you're incorrect.
If you think about it the test is actually pretty useless! You might just as well take a look at each person and say "You don't have the disease". This will be correct 99% of the time and have the same accuracy as if you bother to actually do the test.
So my guess would be 99%. Why would the devisors of the test quote a 99% accuracy figure for a positive test if they knew that that meant you only had a 50% chance of having the disease?
Well done Bry, your go
Cushty, see you about 8pm sharp
; )
So in fact I am 100% right
The common mistake everyone is making is that you are not factoring in the sheer number of false positives there will be in the non-diseased population.
The test is 99% accurate because if you actually have the disease, there is a 99% chance you will be correctly diagnosed. This does not take into account the actual incidence of disease in the population. The reason why it is 50-50 because you have no idea if you have the disease or not and a false positive is just as likely when you factor in both the accuracy and incidence as a correct positive.
So in a sample of 10,000 only 100 people will have it, which means 99 people will have positive test results. But that also means 9900 don't have it, and the inaccuracy of the test means 1% of those - another 99 - will also get positive test results. So from the sample of 198 who had positive results, half of them will actually have it. 50%.
Damn! I studied maths/stats at uni, I hang my head in shame ;-)
EDIT: I also now realise it wasn't you who talked about outcomes, bloody quoting...
I miss the puzzle thread we had a few months ago, definitely some good brain-teasers.
There are 7 pirates on a ship including the captain. They have an order of rank that they all know. So the captain is rank 1 and the least important is rank 7. At any time the pirates can take a vote to remove the Captain, under which circumstances the rank 2 pirate becomes captain. If the vote is tied then the current captain has the deciding vote.
They come across 100 coins (of equal value), the captain must decide how to distribute these coins among the crew. What is the most number of coins he can take for himself without a mutiny?
I should add that the pirates are all mathematical geniuses and completely ruthless with no loyalty whatsoever.
1/100 1/100 99/100 99/100
1/100 99/100 1/100 99/100
1/10,000 99/10,000 99/10,000 9801/10,000
in order: you have the disease, incorrectly diagnosed - you have, correct - you haven't, incorrect - you haven't, correct
99/10,000 is the probability for both positive results - 50% is correct. I'm obviously late to this though