tl; dr - Bất kỳ dự đoán một lần nào từ một người đoán cụ thể đều có thể được giảm xuống một xác suất duy nhất. Tuy nhiên, đó chỉ là trường hợp tầm thường; cấu trúc xác suất có thể có ý nghĩa bất cứ khi nào có một số liên quan theo ngữ cảnh ngoài chỉ một xác suất duy nhất.
Cơ hội để một đồng xu ngẫu nhiên hạ cánh trên Heads là 50%.
Không quan trọng nếu đó là một đồng tiền công bằng hay không; Ít nhất là không phải với tôi. Bởi vì trong khi đồng xu có thể có thành kiến mà một người quan sát có kiến thức có thể sử dụng để đưa ra dự đoán sáng suốt hơn, tôi phải đoán tỷ lệ cược 50%.
Bảng xác suất của tôi là:
Thủ trưởng50 %Đuôi50 %.
But what if I tell someone that the coin has 50% odds, and then they have to make a decision about what happens on two coin flips? Lacking further information, they'd have to default to guessing that coin flips are independent events, arriving at:
SecondflipFirst flipHeadsTailsHeads25%25%Tails25%25%,
from which they might conclude
Same sidetwice50%Headsand Tails50%.
However, the coin flips aren't independent events; they're connected by a common causal agent, describable as the coin's bias.
If we assume a model in which a coin has a constant probability of Heads, PHeads, then it might be more precise to say
HeadsPHeadsTails1−PHeads.
From this, someone might think
SecondflipFirst flipHeadsTailsHeadsP2HeadsPHeads(1−PHeads)TailsPHeads(1−PHeads)(1−PHeads)2,
from which they might conclude
Same sidetwice1−2PHeads(1−PHeads)Headsand Tails2PHeads(1−PHeads).
If I had to guess PHeads, then I'd still go with 50%, so it'd seem like this would reduce to the prior tables.
So it's the same thing, right?
Turns out that the odds of getting two-Heads-or-Tails is always greater than getting one-of-each, except in the special case of a perfectly fair coin. So if you do reduce the table, assuming that the probability itself captures the uncertainty, your predictions would be absurd when extended.
That said, there's no "true" coin flip. We could have all sorts of different flipping methodologies that could yield very different results and apparent biases. So, the idea that there's a consistent value of PHeads would also tend to lead to errors when we construct arguments based on that premise.
So if someone asks me the odds of a coin flip, I wouldn't say ‘‘50%", despite it being my best guess. Instead, I'd probably say ‘‘probably about 50%".
And what I'd be trying to say is roughly:
If I had to make a one-off guess, I'd probably go with about 50%. However, there's further context that you should probably ask me to clarify if it's important.
People often say some event has a 50-60% chance of happening.
If you sat down with them and worked out all of their data, models, etc., you might be able to generate a better number, or, ideally, a better model that'd more robustly capture their predictive ability.
But if you split the difference and just call it 55%, that'd be like assuming PHeads=50% in that you'd basically be running with a quick estimate after having truncated the higher-order aspects of it. Not necessarily a bad tactic for a one-off quick estimate, but it does lose something.