medium Probability
A release-checked medium problem for training Conditional Probability, Discrete Random Variables.
Suppose $N > 0$ is an integer. Let $X_1,X_2,X_3$ be independently selected uniformly at random from the set $\{0,1,\dots,N\}$. Let $p_N = \mathbb{P}[X_1 + X_2 + X_3 = N]$. Find $\displaystyle \lim_{N \rightarrow \infty} Np_N$. If this limit does not exist, enter $-1$.
This Probability problem is tagged Conditional Probability, Discrete Random Variables. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.
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