You go to the doctor for a quick check up. The doctor tells you to take a test on a very rare illness (1 in a million of the population have it, based on experimental and historical data) and so he handles over a test in a box. In the box it says that the test has 99,99% of accuracy.
You take the test and it is positive. So you definitely have that rare condition, right?
What is the actual probability you do?
The problem above it is one of the most counterintuitive I have heard, and at the same time is one of the most fundamental and powerful results you may encounter constantly on your day to day life. It’s the Bayes Theorem.
The (surprising) actual answer to the question above is that in that situation you have around 1% of probabilities of having the disease. Yes 1% (is not a typo). How can that be?
The actual reasoning behind it is not difficult. A test with 99,99% accuracy will (in average) give you 9999 correct results out of every 10.000 tests or 1 false positive every 10.000 tests, and this is the key to understanding Bayes. If the rare condition you are testing is 1 in a 1.000.000 (as in the situation above), what would be the expected results of the test if you test a milion individuals?
The test is expected to give 1 false positive every 10.000 so, you end up with ~100 false positives out of a 1.000.000 tests! That makes 101 positives in total (adding the expected 1 that actually have the disease )
This is: 100 false positives and 1 real positive. So once you know you are one of those 101 positives, your chances of being the real positive is 1 in 101, or ~1%
Unbelievable, right? But how? This is a much more interesting question…
“My dad heard this story on the radio. At Duke University, two students had received A’s in chemistry all semester. But on the night before the final exam, they were partying in another state and didn’t get back to Duke until it was over. Their excuse to the professor was that they had a flat tire, and they asked if they could take a make-up test. The professor agreed, wrote out a test, and sent the two to separate rooms to take it. The first question (on one side of the paper) was worth five points. Then they flipped the paper over and found the second question, worth 95 points: “which tire was it?”
What was the probability that both students would say the same thing?My dad and I think it’s 1 in 16. Is that right?
No, it is not: If the students were lying, the correct probability of their choosing the same answer is 1 in 4”
from “The Drunkard’s Walk: How Randomness Rules Our Lives” by Leonard Mlodinow
What seems counterintuitive or paradoxical about this is, as usually is in probability riddles, confusing the answer from a more “natural” or “common” question with the answer for our actual enquiry.
In this piece we are actually asking “what is the probability that 2 people choose the same random choice from 4 options independently” which is 1/4 , which differs from the more usual question we may wrongly assume like “what is the probability 2 guys choose randomly the correct option out of 4 possible choices” which comes at 1/16 (remember that in the story the boys are lying so there is no actual “correct” answer, any would do).
When stated like this it is easy to see where the key difference lies (and why the answer 1/4 is not paradoxical at all): it lies in the fact that in the first question all 4 choices could be correct (or lead to a successful outcome) while in the second only 1 choice leads to a successful outcome, so in the former no matter which actual choice is made (It only matters that the choices are the same), while in the latter it does (they need to be the same and the correct one on top)
THOUGHT: It is not paradoxical that 2 different questions have 2 different answers!
The Extra Mile
It is interesting to think as well what would be the perception from the point of view of the teacher. He (unlike us, the readers) does not know if the kids are lying or not (that is actual thing he is trying to find out). So, the relevant question for him could be, what is the probability they are lying (or telling the truth) given their answers?
Continue reading “The importance of understanding what’s being asked”