If we have a multinomial distribution $p_1,\ldots,p_n$ and we perform $k$ trials. What is the probability that we see (at least) an outcome $i$ but none of the outcomes is $j$?
2026-04-03 15:47:49.1775231269
Probability of seeing i but not j in k multinomial trials
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This is the probability that none of the outcomes is $j$ minus the probability that none of the outcomes is either $i$ or $j$:
$$ \left(1-p_j\right)^k-\left(1-p_i-p_j\right)^k\;. $$