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I will try to explain what I am trying to ask in my terms because I don't know how to do any better. Feel free to edit for clarity if you understand better than I do what I am trying to ask.

The problem

Say I have a multi-label object $X$ with a total of $n$ possible labels, such that a certain object in that class is defined as

$$ X_{target}(L_1, L_2, L_3,...,L_n) $$

Where any of $L_1$ to $L_n$ can be either $True$ or $False$,

Now say I make a guess, knowing that there are $k$ labels that are $True$ in $X_{target}$, so I choose $k$ labels at random to set to $True$ $$ X_{guess}(L_1, L_2, L_3,...,L_n) $$

What I want to know is what is the probability that, for any $L_i$ in both $X$s, both labels are $True$

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  • $\begingroup$ Are you asking for the probability of a given $L_i$ to be True in both Target and Guess? e.g. Probability of $L_1$ to be True in both? $\endgroup$
    – gunes
    May 2, 2021 at 11:19
  • $\begingroup$ Yes, that is correct. I want to know the probability that it is True in both in the scenario where I necessarily have to choose $k$ entries to be True in both. $\endgroup$ May 2, 2021 at 18:08

1 Answer 1

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Let's assume that your dataset is balanced, i.e. $P(L_i = True)$ is equal for all $i$s. It follows that $P(L_i = True) = k / n$ for $i = 1..n$. The same is true for your guess: let $\hat{L_i}$ be your guesses. Then $P(\hat{L_i} = True) = k/n$ as well. And since your guess was random and does not depend on the target label, it follows that: $$P(\hat{L_i} = True \cap L_i = True) = P(\hat{L_i} = True) P(L_i = True) = (k/n)^2.$$

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  • $\begingroup$ Right, I had arrived at this earlier on but I decided it was wrong because...? I think I was thinking of the probability of choosing a set that contains a given $True$ label. Regardless, you answered the question I posed, thanks! $\endgroup$ May 2, 2021 at 21:34

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