What's The Probability?

For fixed integers \(n\) and \(k\), find all \(k\)-tuples of non-negative integers \( (a_1, a_2, \ldots a_k) \) such that

\[ a_1 + a_2 + \cdots + a_k = n \]

For each integer \( i \) from 0 to \(n\), let \( p_i \) be the probability that one of the \( a_j \) is equal to \(i \). What is the value of

\[ 1\times p_1 + 2 \times p_2 + \cdots n \times p_n ? \]

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