to j165,
It was another (deleted) thread and forum where you mentioned the
probability of getting a head in the fifth throw of a coin after getting 4 consecutive tails.
Allow me to tell you my thoughts, although I'm not an expert in probabilistics, on why one is generally in error when assuming always an a priori probability of 1/2 for all cases.
When looking for a probability in a sequence of events we apply Bayes' theorem of conditional probability, which states that the probability of A happening, given C as a prior event, is P(A/C) = P(A&C)/p(C)
In an unknown family with two children, the four possible time sequences are GG, GB, BB, BG to which we attached the same probability. Knowing that at least one child is a girl (G), P(C)=3/4. To find the probability P(A) of whether the other child is also a girl (GG), we estimate it from P(A&C)=(1/4). Then P(A)=(1/4)/(3/4)=1/3.
Easily seen w/o any theoretical help just from counting the cases.
In short, to estimate the correct probability of an event one has to see the right context, or, in technical jargon, build the right
model.
If one knows beforehand that four throws resulted in tails, you are absolutely right in that the chances for the next being heads is 1/2. The same would happen with the family above, had we said the
first born child was a girl. In this case we are left with two options: GB, and GG, and the chances are again 1/2.
If one knows more about the family or about genetical or hereditary influences, or about the coin's metallurgy or geometry, or of any external influences, barring the chance of its standing on the edge, the conclusion isn't right simply because the
model used isn't right.
If one asked what is the probability of
exactly one head in five throws of a coin the probability is 1/5. The
model would then be composed of: HTTTT, THTTT, TTHTT, TTTHT, TTTTH.
If the question were about the probability of getting "at least" one H, then again the model changes and so does the probability. Can I say: Quod erat demonstrandum ?