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## Homework Statement

Consider a system consists of two subsystem A and A' in which A contains 3 spins and A' contains 2 spins. Suppose that, when the systems A and A' are initially separated from each other, measurements show the total magnetic moment of A to be [itex] - 3{\mu _0}[/itex] and the total magnetic moment of A' to be [itex] + 4{\mu _0}[/itex]. The systems are now placed in thermal contact with each other and are allowed to exchange energy until the final equilibrium situation has been reached. Under these conditions calculate:

(a) The probability P(M) that the total magnetic moment of A assumes any one of its possible values M.

## Homework Equations

Basic statistic and probability tools.

## The Attempt at a Solution

The question implies that each of the spin in A has magnetic moment [itex]{\mu _0}[/itex] while A' has 2[itex]{\mu _0}[/itex].

Originally, the system has the spin arrangement like this:

--- ++

where - means spin down and vice versa.

After they are in contact, they can have the following possibility.

--- ++ [itex] - 3{\mu _0}[/itex]

--+ -+ [itex] - {\mu _0}[/itex]

-+- -+ [itex] - {\mu _0}[/itex]

+-- -+ [itex] - {\mu _0}[/itex]

+-- +- [itex] - {\mu _0}[/itex]

+-+ -- [itex] {\mu _0}[/itex]

++- -- [itex] {\mu _0}[/itex]

--+ +- [itex] - {\mu _0}[/itex]

-++ -- [itex] {\mu _0}[/itex]

-+- +- [itex] - {\mu _0}[/itex]

So [itex]P( - 3{\mu _0}) = \frac{1}{{10}}[/itex], [itex]P( - {\mu _0}) = \frac{6}{{10}}[/itex], [itex]P({\mu _0}) = \frac{3}{{10}}[/itex].

But the answer is [itex]P({\mu _0}) = \frac{6}{7}[/itex], [itex]P( - 3{\mu _0}) = \frac{1}{7}[/itex].

What's wrong?