“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?
If the answers are different, it is 100% clear they are lying (allow me to omit rare conditions that may lead to one of the boys to forget or mistake the correct answer the day after it happened), but what about if they answer the same thing? Obviously it is not 100% sure they are not lying… but is it 75% sure (As the 1/4 -25%- probability above may suggest) ? More? Less?
From the point of view of the teacher we cannot assume before hand that the boys are lying. So we need to account for the option that they were telling the truth, and that, in that case, they would always answer the same choice (the correct one). So that why it is not 3/4 or 75% as We may have thought (the 1/4 probability that we calculated above presupposes that the boys are lying).
So when looking at the whole tree of outcomes we have an initial split 50/50 between the boys lying or telling the truth (we assume the boys are not compulsive lyers or biased in any direction) and we see that the probability (overall) that they both provide the same answer is the sum of the 50% of the cases (telling the truth) plus the 1/4 x 50% (lying but lucky), so a total of 62,5%.
So that’s it then? Is this the number we are looking for? Well not really… That number means that for every 100 times the teacher does this, 62,5 of the times the students will submit the exams with matching answers, but we know as well that only 50 of those 62.5 times they will be genuine truths, so the correct answer is 50 out of 62,5 or 80% of the times.
The teacher then can assume that the boys are telling the truth with 80% confidence if their answers match.
FINAL THOUGHT: when thinking about probabilities, always spend time in understanding (or finding) what is the actual real question you are trying to answer. It makes a difference!