Statistical Physics: Proving "if p(a)=p(b)=p then p(ab) ≤ p^2

In summary, the problem at hand is to prove or disprove the statement "if p(a)=p(b)=p then p(ab) ≤ p^2" for any possible values of p, a, and b. After discussing the definitions of probability, independence, and disjoint events, it was determined that there are two cases to consider: 1) a and b are mutually exclusive, in which case p(a∩b) = 0, and 2) a and b are independent, in which case p(a∩b) = p(a)*p(b) = p^2. However, it is still necessary to prove that this statement holds true for all possible cases and values of p, a, and b
  • #36
rangatudugala said:
how come this possible ?P(a/b) = P(a ∩ b) / P(b)
That's what it is.
 
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  • #37
You have the probability of a given b, so you are only looking at the subset of the possibilities where b has already happened. If the events are independent, the P(a ∩ b)=P(a)P(b), similarly P(a|b) = P(a). Test these against the formula, and you will see how it is true.
 
  • #38
1ba46ff8b3cbad33a7c8eead937c8d34.png


99b11ba7fe4ec33be7a129cf182a32d2.png
 
  • #39
rangatudugala said:
1ba46ff8b3cbad33a7c8eead937c8d34.png


[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.)
 
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  • #40
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"?
 
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  • #41
HallsofIvy 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"?
I hv already answered. probably you didn't see

here rewritten the question
prove or disprove the following
p(a) = P(b) = p (let say some valve, you can use what ever symbol ) then p(a ∩ b) ≤ p^2
 
<h2>1. What is the significance of the equation "p(a)=p(b)=p" in statistical physics?</h2><p>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.</p><h2>2. How does the equation "p(a)=p(b)=p" relate to the concept of microstates and macrostates?</h2><p>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.</p><h2>3. What does the inequality "p(ab) ≤ p^2" tell us about the behavior of systems in equilibrium?</h2><p>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.</p><h2>4. How is the equation "p(a)=p(b)=p" used in the derivation of the Boltzmann distribution?</h2><p>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.</p><h2>5. Can the equation "p(a)=p(b)=p" be applied to all systems in equilibrium?</h2><p>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.</p>

1. What is the significance of the equation "p(a)=p(b)=p" in statistical physics?

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.

2. How does the equation "p(a)=p(b)=p" relate to the concept of microstates and macrostates?

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.

3. What does the inequality "p(ab) ≤ p^2" tell us about the behavior of systems in equilibrium?

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.

4. How is the equation "p(a)=p(b)=p" used in the derivation of the Boltzmann distribution?

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.

5. Can the equation "p(a)=p(b)=p" be applied to all systems in equilibrium?

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.

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