Hmm, Kabbes, I get a different result. I’m no mathematician, but I thought your 0.1% seemed a little low, so I calculated the probability and got 1% instead (still much lower than I expected, but now I understand it). How did you calculate the number you posted?
Let us assume that 10,000 people were randomly~ screened.
Then we can assume for those 10,000 people:
exactly 1 has the disease, who we’ll call Amy
9,999 do not have disease-we’ll call these people Darwin’s hordes
So, when they are screened by the test, Amy tests positive (100% accurate) and 99.99 (=1%*9999) of Darwin’s hordes test positive (these are the 1% false positives). This makes a total of 1 + 99.99, or 100.99 total people who test positive.
Now, your doctor tells you that you are positive. What is the probability that you are Amy, and not just one of Darwin’s hordes?
Well, there were 100.99 people who tested positive, and only 1 of those (Amy) actually has the disease, so the probability that you are Amy is 1/100.99 or 0.00990197, which rounds to 0.99%, or approximately 1%.
I like to do probability problems this way, because it makes them more concrete, and much easier to calculate if you don’t know the abstract rules for a particular case. Of course, that means that I have to tolerate fractional people, but who cares?
~ I think one reason many people overestimate these type of probabilities is that completely random screenings are not that common in our experience.
Most of the time, only people already displaying symptoms or suspecting illness are tested. And because of this, they are much more likely than people in a random screening test to actually have the disease, so the above calculations wouldn’t apply.
Being a member of a risk-group for the disease is another reason many people might get tested. If this disease ran in my family for example, I’m probably more likely to have the disease than the 1% figure I calculated above, since it implicitly assumes I am selected at random from the total population, which I clearly am not.
Yeah, but what are the chances that a guy would actually be thinking about breasts? Large, shapely ones. Moving gently from side to side . . . Uh. Never mind.
MaceMan - you’re right - I’d made the schoolboy error of saying 0.00990197 = 0.1%. :eek:
If you want it on a sounder footing, try this simple formula (the way I did it originally):
P(Y|X) = P(Y|X).P(X)/P(Y) where “X|Y” means “X happens given Y has happened”. This is Bayes’ conditional probability formula.
Let X be the event “person tests positive”
Let Y be the event “person has condition”
Then X|Y is the event “person tests positive given they have the condition” - the probability of this is 100%.
and Y|X is the event “person has condition given that they tested positive” - the event we want to evaluate the probability of.
We already know that event Y has probability 1/10000.
So what is probability X?
Well P(X) = P(X|Y).P(Y) + P(X|¬Y).P(¬Y) where ¬Y = “not Y”
this is because either Y is true or not Y is true.
So P(X) = 100% x 1/10000 + 1% x 9999/10000 = 0.010099
Now we can combine:
P(Y|X) = P(X|Y).P(Y)/P(X)
= 100% x 0.0001 / 0.010099
= 0.00990197
= 1%, give or take the odd 0.1%
Glad we cleared that up.
Point remains though - a probability a lot lower than most would intuitively believe.
Now you must excuse me - I have to go check my last fortnight’s work for similar sloppy errors. Fortunately Excel tends not to make them, however. Which might account for my loss of numerical judgement here, I suppose.
