Quote:
Originally Posted by Snowman
We're not quite saying the same thing. Your math is correct if the question is phrased as "what is the probability that all 6 men are indeed a match for the 6 people Steve thinks they are?"
But that's not quite the same question I was answering above. If this is indeed a Knickerbockers photo, then the subjects in the photo are not independent of one another (independent in the statistical sense). In other words, if one of them is indeed a Knickerbocker, then that increases the likelihood that a second person is also a Knickerbocker. And if 2 are known to be Knickerbockers, then again, it increases the likelihood that a 3rd is, etc. Knickerbockers are likely to be photographed together. So my framing of the question "what are the odds that this is a Knickerbockers photo?" approaches it with that dependence structure in mind. It basically calculates what the odds are of all of his 90% Knickerbocker matches to be wrong rather than what the odds are for each one to be correct independently. My approach allows for, say, 5 of his 6 matches to be correct but him mistaking the identity of the 6th one, thus still making the photo a "Knickerbockers" photo.
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I spoke to my son who is better at math than I am and he started talking about Bayesian probability and I kind of lost him halfway through.
But I think his point was that knowing one person was 100% a Knickerbocker does indeed affect the probability that the others are Knickerbockers…the problem is we don’t know how it affects the probability. For example, if we knew for a fact that this one person
only had his photo taken with other Knickerbockers (and never had a photo taken with family members, business associates, friends, or anyone else, ever) that would affect it in a very positive way…the other people would all have to be Knickerbockers. But if we knew this person was estranged from the other members and refused to be photographed with them, that would affect it in a negative way...the other people could not be Knickerbockers. We would essentially need to know the universe of all the photos this person was in and what percentage of these photos contained only Knickerbockers. Seeing as how we don't know how often/rarely they got photographed together, the odds are still based on these being independent occurrences at 90% per person.
However, if one person was 100% a Knickerbocker, it does help in the fact that the odds have increased from 0.9^6 = 53% to 0.9^5 or 59% that all of them are Knickerbockers.
For the purposes of the above, I am assuming the 90% number to be accurate and that I am correctly conveying what I understood from what my son was telling me.