This might be of interest to people here; it’s an example of a genuine confusion over probability that came up in a friends medical research today. It’s not particularly complicated, but I guess it’s nice to link these things to reality.
My friend is a medical doctor and, as part of a PhD, he is testing peoples sense of smell. He asked if I would take part in a preliminary experiment to help him get to grips with the experimental details.
At the start of the experiment, he places 20 compounds in front of you, 10 of which are type A and 10 of which are type B. You have to select two from that group, smell them, and determine whether they are the same (i.e. both A or both B) or different (one is A, the other B). He’s hoping that people will be able to distinguish these two compounds reliably.
It turned out that I was useless at distinguishing them—over a hundred odd trials I managed to hit 50% correct almost exactly. We then discussed the methodology and realised that it was possible to do a little bit better than 50% without any extra sniffing skills.
Guess they’re different every time. There are more pairs of different compounds from a selection group than pairs of same ones. (For any given compound, there are 9 matches and 10 non-matches.)
More detail in the protocol would be helpful. For example, do you get to repeatedly use the same bottle of the set? If so, I can do the following for a 20 trial set: Pick bottle 1. Now run through the other 19 bottles and guess for each that it is different from bottle 1. I’ll be correct 10 out of 19 trials. This method generalizes in a fairly obvious fashion although it isn’t clear to me if one is going to do n trials whether this is actually the optimal procedure for maximizing how often you are correct. I suspect that one can do better but it isn’t clear to me how.
This might be of interest to people here; it’s an example of a genuine confusion over probability that came up in a friends medical research today. It’s not particularly complicated, but I guess it’s nice to link these things to reality.
My friend is a medical doctor and, as part of a PhD, he is testing peoples sense of smell. He asked if I would take part in a preliminary experiment to help him get to grips with the experimental details.
At the start of the experiment, he places 20 compounds in front of you, 10 of which are type A and 10 of which are type B. You have to select two from that group, smell them, and determine whether they are the same (i.e. both A or both B) or different (one is A, the other B). He’s hoping that people will be able to distinguish these two compounds reliably.
It turned out that I was useless at distinguishing them—over a hundred odd trials I managed to hit 50% correct almost exactly. We then discussed the methodology and realised that it was possible to do a little bit better than 50% without any extra sniffing skills.
Any thoughts on how?
Guess they’re different every time. There are more pairs of different compounds from a selection group than pairs of same ones. (For any given compound, there are 9 matches and 10 non-matches.)
Probability that both compounds are A = P(1st is A)P(2nd is A) = (1/2)(9/19) = 0.24
Probability that both are B = 0.24
Probability that both are same = 0.47
Probability that they are different = 0.53
Conclusion: Always predict they are different.
More detail in the protocol would be helpful. For example, do you get to repeatedly use the same bottle of the set? If so, I can do the following for a 20 trial set: Pick bottle 1. Now run through the other 19 bottles and guess for each that it is different from bottle 1. I’ll be correct 10 out of 19 trials. This method generalizes in a fairly obvious fashion although it isn’t clear to me if one is going to do n trials whether this is actually the optimal procedure for maximizing how often you are correct. I suspect that one can do better but it isn’t clear to me how.
And can the person who downvoted this please explain why they did so?
This sounds like a great applied exercise for Chapter 3 on elementary sampling theory. ;)