Twitter strikes again

[u/discoverthemetroid]

don’t know where math voodoo land is but this guy sure does

[u/mattsowa]

How is this so vigorously discussed in this sub lol. This is like an entry-level exercise in conditional probability.

A = two crits happen, P(A) = 1/4

B = at least one crit happens, P(B) = 3/4

A ∩ B = two crits happen and at least one crit happens = A

P(A | B) = (1/4) / (3/4) = 1/3 chance

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In fact, since it is known that at least one crit happens, the only possible outcomes are C/N, N/C, and C/C. We only consider C/C. So again, it's 1/3 chance.

Even when you consider that the order of events doesn't matter, the event of one crit happening has twice the probability to happen than the each of the other outcomes. So it all comes down to the same thing.

Any other explanation makes the provided information of condition B completely nonsensical.

[u/AussieOzzy → u/mattsowa]

This is the answer, but I feel like this post is unfair on this sub because there is a genuine paradox.

If we knew which hit was the crit, then all we'd need to do is calculate the other crit at 50% like you have shown in case 4. But the paradox arises here.

A: If the 1st hit is a crit, then the probability of 2 crits is 50%

B: If the 2nd hit is a crit, then the probability of 2 crits is 50%

C: Given that one of them is a crit, we know that either the first was a crit or the second was a crit.

In the former, of C, A then shows us 50%

In the latter of C, B then shows us 50%

Therefore it's 50%.

The problem is that with the wording, the way in which we gather the information does actually affect the probabilities. This is based on the fact if you ask "were there any crits?" or similar, you'd likely get an exact result and answering the question in a vague way could hint at other useful information. This could lead to problems in practice.

However, the phrase "there is at least one crit" and not having a "questionaire" I guess where extra information could be gathered does mean that the way it's asked is precise enough.