Story problem...

[46AND2]

Say there is a 45 year old woman who wants to know if she has breast cancer.

Given the following statistics:

1. If she has cancer, there is a 75% chance that cancer will show up on a mammogram

2. If she doesn't have cancer, there is still a 10% chance the mammogram will test positive (a false positive)

3. 1.4% of all women in their 40's have breast cancer

Say the lady goes in and gets a mammogram done, and it comes out positive.

What are the odds that the test is right?

 

[Old Ned]

Oh man I went round in circles on that lol, it's been a long day.

 

[AV1611VET]

Say there is a 45 year old woman who wants to know if she has breast cancer.

Given the following statistics:

1. If she has cancer, there is a 75% chance that cancer will show up on a mammogram

2. If she doesn't have cancer, there is still a 10% chance the mammogram will test positive (a false positive)

3. 1.4% of all women in their 40's have breast cancer

Say the lady goes in and gets a mammogram done, and it comes out positive.

What are the odds that the test is right?

I said 75%, but isn't there a 98.6% chance that I would be wrong?

 

[Paulos23]

Oh no, its Bayesian!

 

[46AND2]

Oh no, its Bayesian!

Was talking about Bayes Theorem with someone the other day, then came across this today, so I'd thought I'd share. ;-)

 

[46AND2]

...but isn't there a 98.6% chance that I would be wrong?

Nope.

 

[46AND2]

Oh man I went round in circles on that lol, it's been a long day.

I know the feeling.

 

[KWCrazy]

There is a 90% chance that the tests are accurate. I Googled it. You never said it wasn't an open book test.

 

[Seipai]

Say there is a 45 year old woman who wants to know if she has breast cancer.

Given the following statistics:

1. If she has cancer, there is a 75% chance that cancer will show up on a mammogram

2. If she doesn't have cancer, there is still a 10% chance the mammogram will test positive (a false positive)

3. 1.4% of all women in their 40's have breast cancer

Say the lady goes in and gets a mammogram done, and it comes out positive.

What are the odds that the test is right?

Hmm, I get 9.5%. Do you need more numbers?

 

[Seipai]

There is a 90% chance that the tests are accurate. I Googled it. You never said it wasn't an open book test.

If you Google incorrectly you will get a wrong answer. It is better to try and figure it out yourself first.

 

[46AND2]

Hmm, I get 9.5%. Do you need more numbers?

Nope. Rounding error?

 

[Seipai]

Nope. Rounding error?

Maybe, I didn't see that there was a poll. I would have chosen your 9.6% answer.

ETA: And now I have.

 

[46AND2]

There is a 90% chance that the tests are accurate. I Googled it. You never said it wasn't an open book test.

I don't know if the given statistics are real. Doesn't matter. It's a hypothetical story problem.

 

[Nithavela]

Say there is a 45 year old woman who wants to know if she has breast cancer.

Given the following statistics:

1. If she has cancer, there is a 75% chance that cancer will show up on a mammogram

2. If she doesn't have cancer, there is still a 10% chance the mammogram will test positive (a false positive)

3. 1.4% of all women in their 40's have breast cancer

Say the lady goes in and gets a mammogram done, and it comes out positive.

What are the odds that the test is right?

Edit: Sorry, misread the question.

 

[46AND2]

The answer is: 9.6%

Here is why:

Say you have a thousand women in their 40s tested. Since 1.4% of women in their 40s have breast cancer, according to statistic 3, 14 of the thousand women, on average, will have cancer.

Of those 14, only 75% will test positive, due to statistic 1. This equals 10.5.

The other 986 women will not have cancer, but an additional 10% of these will test positive, anyway, according to statistic 2. This equals 98.6.

10.5 accurate tests (positive test for those with cancer)
98.6 false tests (positive test for those without cancer)
109.1 total positive tests (accurate + false)

To find the probability that any one test is an accurate positive, we divide:

10.5 (accurate positives)/ 109.1 (total positives)

And that equals 9.6%

 

[Oafman]

60% of the time it works every time