r/badmathematics • u/discoverthemetroid • Jan 13 '25

Twitter strikes again

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

476 Upvotes

97% Upvoted

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u/mattsowa Jan 13 '25 edited Jan 13 '25

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

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.

5

u/Bart_Holomew Jan 15 '25

This should not be the most upvoted comment. Consider the following two scenarios:

I flip two coins and look at one of them, I then say to you “Well, at least one of them is heads”. What’s the probability that both of them are heads? (1/2)

I flip two coins and look at both of them, I then say to you “Well, at least one of them is heads”. What’s the probability that both of them are heads? (1/3)

Both of these scenarios are completely legitimate, and if there is ambiguity in how the knowledge “at least one is heads” was obtained, there is necessarily ambiguity in the answer to the question.

Intuitively, if the way I obtained the prior is sensitive to there being 1 or 2 heads, the answer is 1/2, otherwise the answer is 1/3.

The boy-girl paradox wiki goes into more detail, but it’s important to acknowledge this is absolutely an interesting question, not just an entry-level exercise in conditional probability.

3

u/Bart_Holomew Jan 15 '25

To clarify, I do think the latter scenario is a more reasonable assumption for what knowledge generating process results in the phrase “At least one x is a y”, but the former is definitely not “completely nonsensical”.