# Amazing probability, help me prove it!

"Imagine there is some rare disease that only affects 1 in 10,000 people, but if you get it, it's completely fatal and there is no cure, death is guaranteed. One day you get really paranoid and decide to take a test to see if you have the disease or not, so you go down to your local doctor and he says there's a simple test you can take which is 99% accurate in its diagnosis. So you take the test, then go home and wait. A couple weeks later a letter from the clinic comes through your door and you open it and it says the dreaded words "You have the disease" - so naturally, you would completely freak out, right? However, if you think about it, despite receiving this letter, you still have less than a 1% chance of actually having the disease. Even tho the test is 99% accurate and affects just 1 in 10,000 people."

When I first heard this I couldn't believe it, but after much contemplation it is most definitely true. Imagine if we take a sample size of 1,000,000 people.

As 1 in 10,000 get the disease, this means 100 from our sample size of 1m would have the disease.

Of these 100 people, 99 would be correctly diagnosed as having the disease and 1 would be incorrectly told that they don't have the disease.

There would be 999,900 remaining people who do not have the disease. However, as the test is only 99% accurate, 1% of these 999,900 (i.e. 9,999 people) would receive letters saying they have the disease.

So in total, 9,999 + 99 = 10,098 would be told they have the disease when in fact only 99 of them do. And 99 is less than 1% of 10,098.

So this would seem to be true. However, I brought this up on another forum, and some of the users have brought up some interesting counterpoints. Maybe you could read through the topic and let me know your thoughts. I've said basically everything I can think of to prove it!

mathman
Your description is probably correct. However, you have implicitly assumed a false positive probability as 1% while a false negative is also 1%. In general, there is no reason for both to be the same.

Please can you clarify? I'm not sure what the difference between a false positive probability and a false negative probability is.

disregardthat
The you know that there is 1% chance of the test to be wrong, it IS 1% chance that the test will be wrong. If you calculate that it wouldn't it wouldn't say 1%, would it?

That's just my point of view. Should really the amount of people that it is tested on have any effect on the probability chance of the test?

HallsofIvy
Homework Helper
The you know that there is 1% chance of the test to be wrong, it IS 1% chance that the test will be wrong. If you calculate that it wouldn't it wouldn't say 1%, would it?

That's just my point of view. Should really the amount of people that it is tested on have any effect on the probability chance of the test?
You are assuming that it is correct to say that "there is 1% chance of the test to be wrong." There are two different ways a test can be wrong- it can give a false positive (saying you have the disease when you haven't) or it can give a false negative (saying you do not have the disease when you have). There is no reason to assume those two have the same chance.

You simply changed sets. There's nothing to that.

D H
Staff Emeritus
Halls is correct. There is a very big difference between false positives and false negatives. There is no a-priori reason for these error rates to be equal, nor is there a way to minimize both types of errors simultaneously. BTW, false positives and false negatives are also known as type I and type II errors, respectively.

Initial screening tests for some disease are often designed to have an extremely low false negative rate and a not-so-low false positive rate. In other words, the disease is most likely absent if the test comes up negative. A positive result merely means you have to take some other test again.

The initial post is correct. This is a well-known statistical paradox.

HallsofIvy
Homework Helper
D. H.'s point is a good one- you can reduce the number of "false negatives" by loosening the conditions underwhich you report a positive result (for example if you report a positive result no matter what the result of the test you would NEVER report any negative and so have NO false negatives!) but that invariably increases the number of false positives. A false positive can cause some emotional trauma but a false negative could result in a disease not being treated! Most test procedures accept more false positives in order to reduce the number of false negatives. Yes, if a disease is rare to start with, it is not terribly surprizing, even with a test that has a low percentage of false positives, that most "positive results" sjhould be false. There just aren't enough "true positives".

Question: A wholesalers buys items at £25 each. The wholesalers sells this in batches of 8 to retailers at £36 per item. It is know that, on average , 25 items in a thousand are faulty.

(A) Construct the probability distribution table for the number of faulty items in a batch of 8 items. All probability values must be rounded off to 5 decimal places.

I know i have to do a binomial distirbution for Part (A) lots of thanks to jerry and so can anybody help me with Part B of the same question.

(B) For any faulty item in a batch the wholesaler promises to give 2 extra items, free of further charge, to the retailer who then accepts full responsibility for the batch.
Extend your table of part (A) to indicate the wholesalers profit for each possible value of the number of faulty items in a batch of 8. Extend your table further to calculate the wholesalers expected profit on a batch of 8 items.

Kindly help me in solving the above probability problems, help will be highly appreciated.

Thank you very much in advance.

disregardthat
But doesn't the fact that it can actually give you a positive result even though you do not have the disease even the factors out?

This could be worked with Baysian conditional prob.
Define events:
N=no disease P(N)=1/10000
Y=have disease P(Y)=1-P(N)
Y'=test positive P(Y')=P(Y'/N)*P(N)+P(Y'/Y)*P(Y)
N'=test negitive
WRONG TEST RESULT P(Y'/N) =1/100= P(N'/Y) note: not always equal but assumed here.
GOOD TEST RESULT P(Y'/Y)=.99=P(N'/N)

COMPUTE P(Y')= 1/100*(9999/10000)+.99*(1/10000) =100.98/10000

From Bayes; P(Y/Y')*P(Y')=P(Y'/Y)*P(Y)

We want P(Y/Y')= P(Y'/Y)*P(Y)/P(Y')

=.99*(1/10000)/(100.98/10000)
=.99/100.98=1/102

So less than 1% chance.

HallsofIvy
Homework Helper
Question: A wholesalers buys items at £25 each. The wholesalers sells this in batches of 8 to retailers at £36 per item. It is know that, on average , 25 items in a thousand are faulty.

(A) Construct the probability distribution table for the number of faulty items in a batch of 8 items. All probability values must be rounded off to 5 decimal places.

I know i have to do a binomial distirbution for Part (A) lots of thanks to jerry and so can anybody help me with Part B of the same question.

(B) For any faulty item in a batch the wholesaler promises to give 2 extra items, free of further charge, to the retailer who then accepts full responsibility for the batch.
Extend your table of part (A) to indicate the wholesalers profit for each possible value of the number of faulty items in a batch of 8. Extend your table further to calculate the wholesalers expected profit on a batch of 8 items.

Kindly help me in solving the above probability problems, help will be highly appreciated.

Thank you very much in advance.
PLEASE do not "hijack" another person's thread for new questions that have nothing to do with the orginal question! Start a thread of your own!

HallsofIvy
Homework Helper
But doesn't the fact that it can actually give you a positive result even though you do not have the disease even the factors out?

No- it is quite possible for one probability to be 5% and the other 10%. Those do not "even" out!

mathman
Simple example of difference between falso positive and false negative.

Diabetes type 2 test (overnight fasting)

Current ADA standard <100 not diabetic, 100 - 126 pre-diabetic - >126 diabetic

Older standard < 110 not diabetic, 110 - 140 pre-diabetic, >140 diabetic

As you can see the current test will give less false negatives but more false psoitives as compared to the older test.