Computing the probability of intersection

Abstract Let normal upper Omega 1 comma ellipsis comma normal upper Omega Subscript m Baseline <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> $\Omega _1, \ldots , \Omega _m$ be probability spaces, let bold upper Omega equals normal upper Omega 1 times midline horizontal ellipsis times normal upper Omega Subscript m <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="bold">Ω</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>×</mml:mo> <mml:msub> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> ${\mathbf \Omega }=\Omega _1 \times \cdots \times \Omega _m$ be their product and let upper A 1 comma ellipsis comma upper A Subscript n Baseline subset of bold upper Omega <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>⊂</mml:mo> <mml:mrow> <mml:mrow> <mml:mi mathvariant="bold">Ω</mml:mi> </mml:mrow> </mml:mrow> </mml:math> $A_1, \ldots , A_n \subset {\mathbf \Omega }$ be events. Suppose that each event upper A Subscript i <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> $A_i$ depends on r Subscript i <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> $r_i$ coordinates of a point x element of bold upper Omega <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mrow> <mml:mi mathvariant="bold">Ω</mml:mi> </mml:mrow> </mml:mrow> </mml:math> $x \in {\mathbf \Omega }$ , x equals left parenthesis xi 1 comma ellipsis comma xi Subscript m Baseline right parenthesis <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo>