Is it possible to extra/interpolate probabilities? Is it possible to extra/interpolate probabilities ? Lets say: $\qquad$ $P(X\geq3)=y\qquad$ and $\qquad P(X\leq5)=z$ $\qquad$($y,z$ are already known) Can I calculate $P(X=4)$ and $P(X=6)$? Bonus question: Is it possible to calculate $F(i)=P(X\leq i)$ faster than adding $f(i)+f(i-1)+...+f(0)$?

No, it is not possible.

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In special cases, if your probabilities $y$ and $z$ are equal to $0$ or $1$, it is possible to infer some facts.

For example, if we know that $P(X\geq 3) = 0$ then we know that $P(X =4) = 0$ and $P(X = 6)= 0$.

Or, if you know that $P(X \leq 5) = 1$ then we know that $P(X = 6) = 0$.

Or, if you know that $P(X \leq 5) = 0$ then we know that $P(X = 4) = 0$.