Proof:Pr(E) ≤ 1

From CS2800 wiki
If Pr is a probability measure, then for all events [math]E [/math], [math]\href{/cs2800/wiki/index.php/Pr}{Pr}(E) ≤ 1 [/math]
Proof:
We have


[math]\begin{align*} \href{/cs2800/wiki/index.php/Pr}{Pr}(E) &= \href{/cs2800/wiki/index.php/Pr}{Pr}(\href{/cs2800/wiki/index.php/S}{S} \href{/cs2800/wiki/index.php/%E2%88%96}{∖} (\href{/cs2800/wiki/index.php/S}{S} \href{/cs2800/wiki/index.php/%E2%88%96}{∖} E)) && \href{/cs2800/wiki/index.php?title=Claim:If_A_%E2%8A%86_B_then_A_%3D_B_%E2%88%96_(B_%E2%88%96_A)&action=edit&redlink=1}{\text{since E = S ∖ (S ∖ E)}} \\ &= 1 - \href{/cs2800/wiki/index.php/Pr}{Pr}(\href{/cs2800/wiki/index.php/S}{S} \href{/cs2800/wiki/index.php/%E2%88%96}{∖} E) && \href{/cs2800/wiki/index.php/Claim:Pr(S_%E2%88%96_E)_%3D_1_-_Pr(E)}{\text{using the fact that Pr(S ∖ E) = 1 - Pr(E) with S ∖ E plugged in for E}} \\ &≤ 1 - 0 = 1 && \href{/cs2800/wiki/index.php/Kolmogorov_axiom}{\text{Since Pr(S ∖ E) ≥ 0}} \end{align*} [/math]