Does physics or mathematics allow for a probability of a probability?
As a mathematical theory, modern probability is based on a set of axioms formulated by Kolmogoroff in the 1930's. To make a probability of probability theory, one can see if the axioms make sense in this case.
For instance, if an event has a probability [mu] of probability [nu] of occuring, then can you say in general that the event has a probability [mu][nu] of occuring?
What you are talking about sounds a lot like conditional probabilities.
For events A and B, if:
P(A) = μ
P(B | A) = ν (that's probability that B occurs, given that A occurs)
P(A | B) = 1 (B can only occur if A occurs)
Then we can apply the formula
P(A|B) P(B) = P(B|A) P(A)
P(B) = μ ν
I throw a coin four times, the probailty that I get three heads is 0.25, but you can also say: The probabilty that the probailty after the second throw is 0.0625 is 0.5.
And has Hurkyl says, my example is a conditonal probabilty.
Thanks much for your explanations, folks. I will try to recondition my thinking accordingly.
Could you recommend a simple online source for conditional probability, Hurkyl? The notation slips me.
In the U.S. the typical example of a conditional probability is someone making the second free throw:
Jeff Hornachek (Don't remember spelling) had a 90% free throw rate, so on a double free throw, he had a 81% (or 90% of 90%) chance of making his second shot.
An alternative example would be from statistics or zero knowledge proofs where probability is used as an expression of confidence. Be wary that this type of double probability is something different than the conditional probalitity described above.
For example, there is a 90% probability that that loaded die has a 70% chance of rolling a 6.
Or from polling: There is a 95% probability (expressing confidence in the poll) that each voter has a 45% probability of voting for Arnie.
Sorry, I don't know of any resources in particular... I'd think about any introduction to probability would talk about it though.
'sOK, NateTG gave some excellent examples. Practical interpretations of multiple probabilities tend to elicit different physical variables for each expectation, though. My first free throw might anticipate more rebound action than the second.
A Gaussian curve might be described as an infinite succession of probabilities, whereas a constant statistic could not. Endless deviatives of the Gaussian attest to the potential underlying infinite series of probabilities.
I think you are looking for bayes' Theorum which calculates conditional probabilities. This site explains it pretty well. If you just do a search you'll find lots of info.
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