# Probability of a probability of a . . .

Does physics or mathematics allow for a probability of a probability?

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mathman
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.

mathman,

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?

Hurkyl
Staff Emeritus
Gold Member
What you are talking about sounds a lot like conditional probabilities.

For events A and B, if:

P(A) = &mu;
P(B | A) = &nu; (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)
to get

P(B) = &mu; &nu;

jcsd
Gold Member
Sure:

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.

jcsd
Gold Member
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.

NateTG
Homework Helper
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.

Hurkyl
Staff Emeritus