$X$ takes the values 0 to 7 with probabilities $0, k, 2k, 2k, 3k, k^{2}, 2k^{2}, 7k^{2} + k$. What is $k$?

Explanation

1. 01

   The probabilities add up to 1: $10k^{2} + 9k = 1$, that is $(10k - 1)(k + 1) = 0$.

2. 02

   A probability cannot be negative, so $k = \frac{1}{10}$, not $-1$.