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# Who gets cancer

## Question

This is a relatively easy one from a book i just read so I cant take credit for it:

1% of women get breast cancer. 80% of women who get a positive mamogram result actually have breast cancer. 10% of women who get a negative mamogram result actually have breast cancer.

1)Jane has a positive mamogram result, what is the probability that she has cancer?

2)Mary has a negative mamogram result, what is the probability that she has cancer?

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This is a relatively easy one from a book i just read so I cant take credit for it:

1% of women get female chest appendage cancer. 80% of women who get a positive mamogram result actually have female chest appendage cancer. 10% of women who get a negative mamogram result actually have female chest appendage cancer.

1)Jane has a positive mamogram result, what is the probability that she has cancer?

80%

2)Mary has a negative mamogram result, what is the probability that she has cancer?

10%

Is there something missing from this one?

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Is there something missing from this one?

Unless the answers are something like "Not enough information given" because the questions don't specify breast cancer, I don't think you missed anything.

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Was this question posted the way the author wanted it? I have seen problems like this where you are asked questions like the ones below and you are given the fact that 1% women have breast cancer, and you are given the accuracy of the test, say 75% of the resuts are accurate. Accurate test says definitely yes or no, and the inaccurate tests will read the opposite.

I like it since it is kinda cute that your more likely to not have cancer if you have a positive test.

"Jane goes in for a test, what is the probability that she tests positive and has cancer?"

or

"Mary goes in for a test, what is the probability that she tests positive and does not have cancer?"

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I like it since it is kinda cute that your more likely to not have cancer if you have a positive test.

You're misunderstanding something.

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I like it since it is kinda cute that your more likely to not have cancer if you have a positive test.

You're misunderstanding something.

Woops, let me try to clear this up. Lets look at a 10,000 women.

They all get a mamogram test that is 75% accurate. Of the healthy 9,900(10,000*.99)women, 7,425(9,900*.75) women will get a correct result saying cancer cancer is not present and 2,475(9,900*.25) will get an incorrect result saying cancer is present. Of the 100(10,000*.01) women who do have cancer, 75(100*.75) will get a correct result saying cancer is present and 25(100*.25) will get an incorrect result saying cancer is not present.

So, out of our total women 2,550(2,475+75) who received a positive result for cancer, only 75 truely have cancer. So if a woman tests positive for cancer, she is roughly 3%(75/2,550) likely to have cancer. On the other hand, out of our total women 7,450(7,425+25) who received a negative result for cancer, 25 will actually have cancer. So if a woman tests negative for cancer, she is roughly .3%(25/7,450) likely to have cancer.

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They all get a mammogram test that is 75% accurate.

There is no way to determine that. We must determine that the 1% of women who get breast cancer don't reflect the same amount of women who get mammograms that get breast cancer. Otherwise, because of the stats given in the riddle, the numbers don't add up.

So if a woman tests positive for cancer, she is roughly 3%(75/2,550) likely to have cancer.

No math is needed to determine that you're wrong. The riddle states "80% of women who get a positive mammogram result actually have breast cancer."

So if a woman tests negative for cancer, she is roughly .3%(25/7,450) likely to have cancer.

Same here: "10% of women who get a negative mammogram result actually have breast cancer."

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I guess I shouldn't have posted this. In my dialogue earlier, I mentioned a similar problem to the one posted. You replied that I was misunderstanding something, so I worked out the problem that I posted instead of the one the author posted. I was confused by your response, so I just worked it out. What I posted is correct given the example that I presented. Sorry for the confusion.

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I suppose there is a mistake in the question. If + results have %80 probability, - results have %10 probability, then in the population at least %10 of women must have breast cancer. Assume that there is no + result, all are -, then %10 of those women has breast cancer. If it is not mamogram but breast biopsi, then it may be true, because only suspected women go to biopsi, and a %10 false negative result is logical when total rate is %1. But today nearly every woman goes to mamogram after 45 years old. According to the question less then % 10 of women go to mamogram and this is not logical even in my 3. world county.

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Lets look at a 10,000 women.

They all get a mamogram test that is 75% accurate. Of the healthy 9,900(10,000*.99)women, 7,425(9,900*.75) women will get a correct result saying cancer cancer is not present and 2,475(9,900*.25) will get an incorrect result saying cancer is present. Of the 100(10,000*.01) women who do have cancer, 75(100*.75) will get a correct result saying cancer is present and 25(100*.25) will get an incorrect result saying cancer is not present.

So, out of our total women 2,550(2,475+75) who received a positive result for cancer, only 75 truely have cancer. So if a woman tests positive for cancer, she is roughly 3%(75/2,550) likely to have cancer. On the other hand, out of our total women 7,450(7,425+25) who received a negative result for cancer, 25 will actually have cancer. So if a woman tests negative for cancer, she is roughly .3%(25/7,450) likely to have cancer.

I had something like this puzzle for an interview to get into oxford university... they also tacked on the end:

What could they do to reduce the possibility of false positives. The answer I gave (and they expected) was 'take the test twice.' But I thinik that in the real world there would be a good chance that the failure is sytematic, not probabilistic, so a person that tests false positive once, probably has something different in their body chemistry that would always create false positives.

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I had something like this puzzle for an interview to get into oxford university... they also tacked on the end:

What could they do to reduce the possibility of false positives. The answer I gave (and they expected) was 'take the test twice.' But I thinik that in the real world there would be a good chance that the failure is sytematic, not probabilistic, so a person that tests false positive once, probably has something different in their body chemistry that would always create false positives.

Taking the test twice usually reduces false positives in biochemical tests. But for mamogram it is not so, because the radiolog looks an image and decide if the patient has cancer or not. Some tissue structures may resemble cancer, and if he takes a second mamogrami these tissues will continue to resemble cancer. If the radiolog tries to interpretate these tissues as normal, then he will miss many real cancer cases, and then false negatives will increase.

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