I know about multinoulli distribution, but I found a different explanation in a book that I have been reading and I didn't quite get it. It says:
The multinoulli, or categorical, distribution is a distribution over a single discrete variable with $k$ different states, where k is finite. The multinoulli distribution is parameterized by a vector $p ∈[0,1]^𝑘-1$, where $p_i$ gives the probability of the $i$-th state. The final, $k$-th state's probability is given by $1−(1^𝑇)\cdot 𝐩$. Note that we must constrain (1^𝑇)𝐩 ≤ 1.
I didn't understand how this represents the final state like this $1−(1^𝑇)\cdot 𝐩$ and anything about the vector.
If someone can provide detailed explanation it will be a real help.