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**Iconoclast**View PostNo, I got what you were getting at but just was pointing out the probability of things in real life are rarely as simple as that. But the main point is that you're still focusing on probabilities that do not matter with regards to the question of whether or not there's a connection between the floorboards and the phone call. At some points you say you're not making that claim, but as I've indicated, you continually are making statements that show otherwise. It's those statements that I'm focused upon because those are where the important error is, the quibbles about whether or not to divide the repair probabilities as a flat or increasing distribution is not really important, it's whether or not those probabilities are at all informative with regards to the diary/floorboards/phone call connection.

Now, you asked if there's a simple scenario where we can calculate probabilities, etc. Of course there is, we can simply set up such a safe example, which you did before, flipping a fair coin. This is why coin flips, or die rolls, are often used in statistics lectures. They are nice, closed, simple situations. And, we also define out any annoying real world complications (like edge landing, etc) and talk about our theoretical "fair coins". We define it out of reality to get at the underlying notion of probability theory.

So, if I flip a fair coin, it has by definition a 50% chance for heads and 50% chance for tails.

If I flip a coin 5 times, and get all heads, I have 1/2^5 probability of getting that, or 3.125% chance probability for a fair coin.

So, if I saw that outcome, I might then question if indeed my coin is fair, I might start to think it is biased. So I flip it again, and get a head.

Have I tested the coin's fairness? Do I calculate that 2^6 = 1.5625% and present that as evidence my coin is biased?

If you said yes, you've got that answer wrong on the exam.

See, once I had my 5 heads a fair coin gives me a 50% chance of getting a head on that critical test throw. And, I only flipped it a sixth time because I happened to observe the first 5 heads. Without them, I wouldn't have thought my coin biased. It was that observation, which is admittedly rare, that made me formulate a hypothesis about bias in the first place. Without those known events, the hypothesis would not exist because it is a post-hoc one offered only as a possible explanation for what I have observed. The question itself might be a rare event!

That means I can't use those 5 flips as part of the test because they are the observations that made me formulate the hypothesis; my hypothesis only exists because of those known events (I know I'm being repetitive, but this is what I was getting at earlier when I was mentioning things like how known events complicate matters, and counter-intuitive - we want to use those 5 flips as evidence, it feels like we should, common sense tells us we should - , and so forth, and all those things telling us to use them are wrong).

See the past 5 heads happened, and while they might be rare for a fair coin they can still happen. I'm only considering the hypothesis that the coin is biased because it's been observed, the question itself only exists as a result of those known events. But hypotheses create predictions for future events that I do not yet know, and those I can test to see if they also happen. I now have to collect new evidence to test my hypothesis, which may only exist by chance, and one single flip has a 50% chance of being heads if my coin is fair. So no, my "6th flip" is not enough, so I do 5 more flips and if in those next 5 flips I get 3 tails and 2 heads, I'm probably going to realize my coin was fair all along and that unlikely observation that made me question it's fairness in the first place was just one of the rare events that can happen. But if my hypothesis were correct, then my next pattern of flips will also produce evidence that is unlikely for a fair coin to produce (because my hypothesis is that the coin is biased to come up heads and if that is true, I will continue to get more heads than tails).

That's a simple example that parallels exactly what we're dealing with. An observation has been made, the two events (our 5 heads). We formulated a hypothesis involving a connection between them (biased coin, common event), with the boring alternative that they co-occurred by chance (fair coin, rare event). Without those two events being already known (floorboards & phone call) though, we would not have formed the hypothesis that the diary was under the floorboards, etc. We aren't talking about the diary coming from amid the plumbing, for example, because we hadn't observed that plumbing work was done at Battlecrease. There are lots of other hypotheses that do not "exist" in the discussion, what we want to know is whether or not this one "exists" simply because it was the particular rare event that happened (i.e. it is just as unlikely to get HTHTH, as HHHHH; but some pattern has to arise, and each one is just as rare as the others).

We do not, and cannot, use the initial probability of the specific events to test the hypothesis because we only have the hypothesis because they are known events - they are the question, they are not the answer. You're trying to rephrase the question in the form of an answer when you reuse them but this is not reverse Jeopardy.

Note, as I say, I'm offering no opinion on the diary itself, I'm only getting at the use of the probabilities being discussed here, and the ones being discussed are irrelevant with regards to whether or not the diary came from under the floorboards. They are known events, they are what created the hypothesis, they are not involved in the answer to the question they create.

- Jeff

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