Find a good supremum as you can for probability $P (X≤-1)$

Question

Let $X$ be a random variable whose possible set of values ​​is: $$Ω_X = \{- 5, -4, -3, -2, -1,0,1,2,…, 10\}$$

In addition it is given that $E(X) = 6$.

Find a good supremum as you can for probability $P(X≤-1)$.

Answer 1

Possible solution. Only values with non-zero probabilities are $-1$ and $10$. To get $E(X)=6$, set $P(10)=\frac{7}{11}$ and $P(-1)=\frac{4}{11}$, so sup for $P(X\le -1)=\frac{4}{11}$.