Neither student was right or wrong. This is a Bayesian problem. Each student had a distribution of priors based on evidence. Each source of evidence had its own level of quality which stipulated the variance of the two sets of priors. The challenge is how to combine these priors to create a posterior distribution.

Another way to look at it is that, because "random variation" is how we refer to unknown influences, the perfect-knowledge probability was either 1 or zero. Neither student had perfect knowledge.

> [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

Which is then followed by:

> Why Constructing a Viable Argument and Making Sense of the Reasoning of Others is Crucial. This may seriously be the most vital mathematical practice.

Where the emphasis is on developing an understanding, a viable one (ostensibly leading to a correct answer).

Where the emphasis is on discussion and understanding each others feelings.

>We brought it back together as a whole class to follow it up and each group shared the most interesting conversation that they had.

Where's the part about explaining a correct methodology and the correct answer? I don't believe that rote memorization is the right way to teach math, but discussion and actual understanding are not the same thing. Discussion can LEAD TO understanding, but should not be mistaken for understanding.

I disagree: The teacher's job is to teach mathematical reasoning, not to teach students to retain meteorological trivia from earlier in the day.

That's why I hope the teach corrected this student:

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

His or her mathematical reasoning was wrong; having two options does not mean there is a 50-50 chance.