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But being shown a goat can change the probability. If Monty picks randomly (which is not the original problem but is interesting to consider) then there is no benefit to switching.

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Very true! It depends on the particular variation[1] of the game being played. Which is one difficulty of MH - the ambiguity. Probabilities change when the initial conditions are slightly different or when the problem is interpreted differently. I've never assumed Monty picks at random. But it's entirely possible someone else would make that assumption, in turn changing the optimal strategy.

[1] https://en.wikipedia.org/wiki/Monty_Hall_problem#Variants

Monty picking randomly adds a third outcome. You then have

a) Staying wins (Monty opens to a goat) b) Swap wins (Monty opens to a goat) c) First pick lost, no chance to swap (Monty opens to the car)

each of equal probability.

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