> [One student] though the probability was 50% since it would either rain or not.

That student might have the right answer, but their explanation is dead wrong. I hope that (common) error was pointed out to that student and the whole class. The story doesn't say that it was.

I should not have said the other student was "right." Their answer is just rote repetition of something they heard, without an understanding of the methodology, but their answer is more likely to be correct.

If, in the end, they decided to average the two answers... then their discussion was unproductive and did not arrive at the correct conclusion.

You may be right about that. Unless I have flubbed my calculations, if P(estimated chance of rain=.9|rain) = 1.4(P(rain)), then Bayes theorem leaves the updating at average(.5,.9). Also, an interesting reference about selecting priors: see 2.2 in http://www.stat.cmu.edu/~kass/papers/rules.pdf. Which is not to say that every set of procedures to calculate answers will produce equal errors. There are provably better procedures and provably better answers, given a set of assumptions.

What exactly is the correct conclusion for them?

The right answer here is not a single correct number, but a handful of correct technique or methodology. Counting the number of rainy days in the last x days would be one possible method of prediction. There are many wrong ones too, like "rain has four letters, so it's 4%" or "there are two choices, so it's 50%."

I hope the teacher taught the students which answers were right and which were wrong. If they did then I retract my criticism, even though that wasn't clear in the article.

Technically, lacking in any evidence one way or the other, 50% is a correct prior. There were two mutually exclusive possibilities, the laws of probability demand all mutually exclusive propositions add to 100%, and we become subject to various biases if we refuse to assign uniform priors in this case.

Source: Probability Theory by E.T. Jaynes