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RUber
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That's what it is.rangatudugala said:how come this possible ?P(a/b) = P(a ∩ b) / P(b)
That's what it is.rangatudugala said:how come this possible ?P(a/b) = P(a ∩ b) / P(b)
rangatudugala said:
[PLAIN]https://upload.wikimedia.org/math/9/9/b/99b11ba7fe4ec33be7a129cf182a32d2.png[/QUOTE] [Broken]
Your formula ##P(A \cap B) = P(A) \, P(B)## is NOT a general, true formula; it is true if and only if ##A## and ##B## are "independent" events, such as getting 'heads' on toss 1 of a coin and getting 'tails' on toss 2, or successive particle emissions from a radioactive element. There are millions of real-world examples where it is false. (Or, maybe I mis-read the intent of your post, in which case you might have left out some crucial clarifying information.)
I hv already answered. probably you didn't seeHallsofIvy said:Do you not know that "P(a or b)= P(a)+ P(b)- P(a and b)"?
Also, way back in post #5, Ruber asked what "ab" meant and you never answered. Is it "a and b" or "a or b"?
The equation "p(a)=p(b)=p" represents the principle of detailed balance in statistical physics. This means that the probability of a system transitioning from state A to state B is equal to the probability of transitioning from state B to state A. This principle is important in understanding the behavior of systems in equilibrium.
In statistical physics, a microstate refers to the specific configuration of a system's particles, while a macrostate refers to the overall properties of the system. The equation "p(a)=p(b)=p" implies that the probability of a system being in a specific microstate is equal to the probability of being in any other microstate that corresponds to the same macrostate. This is known as the principle of equal a priori probabilities.
The inequality "p(ab) ≤ p^2" is known as the Gibbs inequality and is a fundamental result in statistical physics. It states that the joint probability of two events occurring is always less than or equal to the product of their individual probabilities. This inequality is important in understanding the stability and behavior of systems in equilibrium.
The Boltzmann distribution is a probability distribution that describes the distribution of particles in a system at equilibrium. The equation "p(a)=p(b)=p" is used in the derivation of this distribution by assuming that all microstates are equally probable, which leads to the Boltzmann distribution.
The equation "p(a)=p(b)=p" can be applied to systems in thermal equilibrium, where the temperature is constant. However, it may not be applicable to other types of equilibrium, such as chemical or mechanical equilibrium, where different principles and equations may apply.