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Given question below is a slightly modified cliche example for probability.

The Hat Problem. Suppose that 4 people throw their hats in a box and then each picks one hat at random. (Each hat can be picked by only one person.) What is the expected value of X, the number of people that get back their own hat?

For n = 4, we create a sample space whether people gets his/her hat or not. After we create sample space through permutations, we derive our probability mass function by using discrete uniform probability law. Then we can solve it for:

$E[X] = \sum\limits_{x} x*p_{x}(x)$

Since we assume all permutations are equally likely, is this solution can be accepted as valid? How to show equally likeliness for permutations?

If the solution doesn't valid, what is the proper way to solve without knowing premise of equally likeliness for permutations?

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The language is subtle, but the phrase "at random" would be intended to convey a uniform probability in this case. I'd go further to argue that in most cases, the phrase "at random" is used to imply either a discrete or continuous uniform distribution, depending on the context.

It would not be possible to prove equal likelihood in a question like this; it must be baked into the assumptions. The answer to your implicit question of: "If the distribution was not uniform, could it affect the value of $E[X]$?" is yes, which is another context clue in favor of choosing a uniform distribution here.

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