A display demonstrating probability consists of a board full of pegs with four bins at the bottom. A ball is placed at the top and bounces through the pegs before ending up in one of the bins. Each bin can hold multiple balls. The balls are distinguishable. (a) How many microstates are possible in a macrostate with two balls? (b) How many microstates are possible in a macrostate with two balls, where at least one of the balls lands in the first bin? (c) How many microstates are possible in a macrostate with two balls, where exactly one of the balls lands in the first bin? (d) What is the entropy, in units of the Boltzmann constant, of the macrostate from part (a)? (e) What is the entropy, in units of the Boltzmann constant, of the macrostate from part (b)? (f) What is the entropy, in units of the Boltzmann constant, of the macrostate from part (c)?
A display demonstrating probability consists of a board full of pegs with four bins at the bottom. A ball is placed at the top and bounces through the pegs before ending up in one of the bins. Each bin can hold multiple balls. The balls are distinguishable.
(a) How many microstates are possible in a macrostate with two balls?
(b) How many microstates are possible in a macrostate with two balls, where at least one of the balls lands in the first bin?
(c) How many microstates are possible in a macrostate with two balls, where exactly one of the balls lands in the first bin?
(d) What is the entropy, in units of the Boltzmann constant, of the macrostate from part (a)?
(e) What is the entropy, in units of the Boltzmann constant, of the macrostate from part (b)?
(f) What is the entropy, in units of the Boltzmann constant, of the macrostate from part (c)?
(g) Assume the probability of a ball landing in any given bin is the same for all bins. If we have a macrostate with two balls, what is the probability at least one of the balls is in the first bin?
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