What probability rule is this?
If $U_1$, $U_2$, are $U(0,1)$ distributed, then $$P(U_1>u,U_2>u) = 1-P(U_1\leq u)-P(U_2\leq u) + P(U_1\leq u,U_2 \leq u)$$
I don't understand the intermediate steps that lead up to this.
2026-04-03 11:07:26.1775214446
What probability rule is this? $P(U_1>u,U_2>u) = 1-P(U_1\leq u)-P(U_2\leq u) + P(U_1\leq u,U_2 \leq u)$
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$$\{U_1>u,U_2>u\} = \{U_1>u\}\cap \{U_2>u\}$$ Then the opposite event is $$\{U_1>u,U_2>u\}^c = \{U_1\leq u\}\cup \{U_2\leq u\}$$ Probability of the opposite event is $$ \mathbb P(\{U_1>u,U_2>u\}^c) = \mathbb P(\{U_1\leq u\}\cup \{U_2\leq u\}) $$ $$ = \mathbb P(U_1\leq u) + \mathbb P(U_2\leq u) - \mathbb P(U_1\leq u, U_2\leq u). $$ since $$\mathbb P(A\cup B)=\mathbb P(A)+\mathbb P(B) - \mathbb P(A\cap B).$$ And $$ \mathbb P(U_1>u,U_2>u) = 1- \mathbb P(\{U_1>u,U_2>u\}^c). $$