= Failure-probability budget
{title2=$\mathbb P(\bigcap_iF_i^c)\geq\max\{0,1-\sum_i\delta_i\}$}
A <failure-probability budget> assigns nonnegative numbers $\delta_i$ to a finite or <countable> family of failure <events> $F_i$ in a common <probability space>, with verified bounds $\mathbb P(F_i)\leq\delta_i$. The <union bound> guarantees that the <probability> of any failure is at most $\min\{1,\sum_i\delta_i\}$. By <De Morgan's laws> and the <complement rule of probability>, the <probability> that every requirement succeeds obeys
$$
\mathbb P\left(\bigcap_i F_i^c\right)=1-\mathbb P\left(\bigcup_i F_i\right)\geq\max\left\{0,1-\sum_i\delta_i\right\}.
$$
Thus $\sum_i\delta_i\leq\delta$ guarantees simultaneous success with <probability> at least $1-\delta$, even when all failure <events> depend on the same observations. The numbers $\delta_i$ are upper allowances; adding them is not a calculation of the exact failure <probability>.
For three requirements, give each a <failure-probability budget> $\delta_i=1/20$. The total budget is $3/20$, so simultaneous success has <probability> at least $17/20$. On a <sample space> of twenty equally likely outcomes, let $F_1=\{1\}$, $F_2=\{2\}$, and $F_3=\{3\}$. These <disjoint events> attain the budget and the success bound. If instead all three failure <events> equal $\{1\}$, their individual <probabilities> are unchanged, but exact failure has <probability> $1/20$ and simultaneous success has <probability> $19/20$. This second model shows how overlap can leave the guarantee conservative.
\Image[/union-bound-failure-budget-sample-space.png]
{title=A shared <failure-probability budget> covers both separate failures and one common failure}
{description=The two panels each have twenty equally likely numbered outcomes. The blue, orange, and purple marks identify membership in $F_1$, $F_2$, and $F_3$, respectively. Their individual <probabilities> are $1/20$ in both panels. The <union> has three outcomes on the left and one on the right, while the <union bound> is $3/20$ in both. The <event> of simultaneous success is the <complement of an event> of any failure and contains seventeen or nineteen outcomes. These are illustrative <probability> models, not measured failure rates.}
For $n$ requirements, an equal allocation $\delta_i=\delta/n$ is convenient; unequal allocations also work when some requirements are harder to satisfy. For infinitely many requirements indexed by $i\geq1$, the allocation $\delta_i=\delta/2^i$ uses total budget $\delta$ by the <geometric series>. A fixed positive allowance for every requirement instead gives an infinite sum and no positive simultaneous-success guarantee from the <union bound>. The individual <probability> bounds must cover all the failure <events> being combined: omitting possible failure modes leaves the resulting guarantee about the listed requirements only. The same addition of error allowances underlies the <Bonferroni correction> in simultaneous <hypothesis tests>.
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