Probability of a probability of a. .

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In summary, according to mathematics, modern probability is based on a set of axioms that make sense in this case. Probability theory can be used to calculate conditional probabilities, which is what you are asking about.
  • #1
Loren Booda
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Does physics or mathematics allow for a probability of a probability?
 
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  • #2
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.
 
  • #3
mathman,

For instance, if an event has a probability [mu] of probability [nu] of occurring, then can you say in general that the event has a probability [mu][nu] of occurring?
 
  • #4
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)
to get

P(B) = μ ν
 
  • #5
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.
 
  • #6
And has Hurkyl says, my example is a conditonal probabilty.
 
  • #7
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.
 
  • #8
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.
 
  • #9
Sorry, I don't know of any resources in particular... I'd think about any introduction to probability would talk about it though.
 
  • #10
'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.
 
  • #11

FAQ: Probability of a probability of a. .

1. What is the definition of "probability of a probability of a"?

The probability of a probability of a refers to the likelihood of a certain event or outcome occurring, given that another event or outcome has already occurred. It is a measure of the uncertainty of an event that is dependent on another event.

2. How is the probability of a probability of a calculated?

The probability of a probability of a can be calculated using the formula P(A|B) = P(A∩B) / P(B), where P(A|B) represents the conditional probability of event A given that event B has occurred, P(A∩B) represents the joint probability of both events occurring, and P(B) represents the probability of event B occurring.

3. What is the difference between a regular probability and a probability of a probability?

A regular probability refers to the likelihood of a single event occurring, while a probability of a probability takes into account the likelihood of that event occurring given that another event has already occurred. In other words, a probability of a probability is a conditional probability that considers the relationship between two events.

4. Can the probability of a probability of a be greater than 1?

No, the probability of a probability of a cannot be greater than 1. This is because the probability of an event occurring cannot be greater than the probability of that event given that another event has already occurred. In other words, the probability of a probability is always equal to or less than the probability of the initial event.

5. How can the probability of a probability of a be useful in real-world situations?

The probability of a probability of a can be useful in predicting the likelihood of complex events or outcomes, where one event is dependent on another. It can also help in decision making, as it provides a more accurate understanding of the uncertainty involved in certain situations. For example, in medical diagnosis, the probability of a certain disease given a specific set of symptoms is a probability of a probability.

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