Test your knowledge of Probability

[ruff_hi]

Imagine that you are a physician and a 45-year-old patient of yours just had a positive mammogram. You also know the following facts:

• There is a 0.8 percent probability that a 40-to-50-year-old woman has breast cancer;
• If a woman has breast cancer, there is a 90 percent probability that she will have a positive mammogram;
• If a woman does not have breast cancer, there is a 7 percent probability that she will have a false positive mammogram.

What do you tell your patient is the probability that she actually has breast cancer?

[ruff_hi]

If your answer isn't on the list, please post your answer in the thread. A free cookie to the first correct answer.

[JujuLautre]

I say 10% approximately

[ori]

I agree with JujuLautre

Spoiler for why it is 9.4%:

800 out of 100 000 women of that range would have BC, of these 720 would have a true positive mammogram
99200 out of 100 000 women would not have BC, of these 6944 would have a false positive mammogram
-> the chance now for a true positive out of all positives is 720/(6944+720) *100 or 720/7664 *100 or 9.4%

Moderator Action: please do make clear at some point soon, why this should stay in civ4 forums...

[AJ11]

7.664%

Spoiler:

Working it out:
There is a 0.8% chance she will have cancer, or a 0.008 probability.
There is a 99.2% chance that she will NOT have cancer, or a 0.992 probability.

If she has, there is a 90% chance of returning a positive, or 0.9 probability.
If she does not, there is a 7% chance of returning a positive, or a 0.07 probability.

Thus, the chance of you saying "yes" will be:
probability of cancer * probability of positive + probability of no cancer * probability of positive
In numbers: (0.008 * 0.9) + (0.992 * 0.07) = 0.0072 + 0.06944 = 0.07664 or 7.664%

EDIT: We had different interpretations of what he said :P

[uat2d]

I voted ~90%, but I was wrong when I did so. It's 63%.

This is what we know:
P(cancer) = .008
P(not cancer) = .992
P(positive mammogram|cancer) = .9
P(not cancer|positive mammogram) = .07

P(pm|c) = P(pm and c)/P(c) <=> P(pm and c) = 0.0072
P(nc and pm) = P(c)-P(c and pm) <=> P(nc and m) = 0.0008
P(nc|pm) = P(nc and pm)/P(pm) <=> P(pm) = 2/175

P(c|pm) = P(c and pm)/P(pm) <=> P(c|pm) = .63

Spoiler:

P(x|y) means probability of x knowing that y
c = cancer
nc = not cancer
pm = positive mammogram

[ori]

I could change your vote if you care about that...

[uat2d]

Originally Posted by ori

I could change your vote if you care about that...

If you don't mind, I'd appreciate it. Not that it matters much, but yeah, it would be kind of you.

[babar]

I must be terrible at probabilities, because I just can't see past 93%. (7% chance a positive result will be wrong, with the rest being irrelevant).

[ParadigmShifter]

It's just Bayes' Theorem

http://en.wikipedia.org/wiki/Bayes'_theorem

[mintegar]

my gut feeling is 0.8%, since just because of the positive test there is no way of knowing if it came from the (7% of 99.2%) of healthy women where the test comes back wrong, or from the (90% of 0.8%) of women with cancer.

[uat2d]

Originally Posted by ParadigmShifter

It's just Bayes' Theorem

http://en.wikipedia.org/wiki/Bayes'_theorem

I never knew about that! It can make these kind of exercises so much more simple!

Not that I need it now for this exercise, but I'll add it to my calculator right away!

[Kid R]

We have the results back from an apparently quite accurate test - it's not looking good for her at all.

[TheMulattoMaker]

I'd flip a coin, ignore the results, and go get a cup of coffee. Everybody be real happy that I work on electronics instead of people.

[ruff_hi]

Originally Posted by ori

Moderator Action: please do make clear at some point soon, why this should stay in civ4 forums...

because the understanding of probability and how it is sometimes misunderstood is fundamental to the combat engine underlying civ4. If people understood probability better, then their civ4 combat experience would be a happier event ...

     Aggg - I just lost 3 95% victory battles in a row - that cannot happen!

[ruff_hi]

Now, on the flip side ... Imagine the above woman got a negative mammogram. Using the same information as above, what is the probability that she truly doesn't have breast cancer?

[r_rolo1]

@ post 15

Well, ruff, that is another issue OFC that people misunderstand odds a lot, but the example you are posting in the last post has some reasons ... linked to the fact that the RNG used in this game is quite more prone to rows of good/bad results than a actual random source ... the individual results are random enough, though ...

[martix]

Also, PRNG's(like the one in the game) tend to exhibit anomalies on range boundaries.
Meaning any comparisons with boundary conditions is gonna skew results even more.

[uat2d]

Originally Posted by ruff_hi

because the understanding of probability and how it is sometimes misunderstood is fundamental to the combat engine underlying civ4. If people understood probability better, then their civ4 combat experience would be a happier event ...

Aggg - I just lost 3 95% victory battles in a row - that cannot happen!

Of course it can't happen, except on the 0.0125% of times it happens.

In fact, in only 14 battles with 95% odds, there's less than 50% chances you'll lose at least once. That cannot happen!

[ori]

Originally Posted by ruff_hi

Now, on the flip side ... Imagine the above woman got a negative mammogram. Using the same information as above, what is the probability that she truly doesn't have breast cancer?

99.9% instead of the 99.2% she had prior to the test...