exercise:F65ebed7a2: Difference between revisions

Latest revision as of 01:04, 13 June 2024

Let [math]A_1[/math], [math]A_2[/math], and [math]A_3[/math] be events, and let [math]B_i[/math] represent either [math]A_i[/math] or its complement [math]\tilde A_i[/math]. Then there are eight possible choices for the triple [math](B_1, B_2, B_3)[/math]. Prove that the events [math]A_1[/math], [math]A_2[/math], [math]A_3[/math] are independent if and only if

[[math]] P(B_1 \cap B_2 \cap B_3) = P(B_1)P(B_2)P(B_3)\ , [[/math]]

for all eight of the possible choices for the triple [math](B_1, B_2, B_3)[/math].

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-<div class="d-none"><math>
+Let <math>A_1</math>, <math>A_2</math>, and <math>A_3</math> be events, and let <math>B_i</math> represent either <math>A_i</math> or its complement <math>\tilde A_i</math>.  Then there are eight possible choices for the triple <math>(B_1, B_2, B_3)</math>.  Prove that the events <math>A_1</math>, <math>A_2</math>, <math>A_3</math> are independent if and
-\newcommand{\NA}{{\rm NA}}
-\newcommand{\mat}[1]{{\bf#1}}
-\newcommand{\exref}[1]{\ref{##1}}
-\newcommand{\secstoprocess}{\all}
-\newcommand{\NA}{{\rm NA}}
-\newcommand{\mathds}{\mathbb}</math></div> Let <math>A_1</math>, <math>A_2</math>, and <math>A_3</math> be events, and let <math>B_i</math> represent
-either <math>A_i</math> or its complement <math>\tilde A_i</math>.  Then there are eight possible choices for the
-triple <math>(B_1, B_2, B_3)</math>.  Prove that the events <math>A_1</math>, <math>A_2</math>, <math>A_3</math> are independent if and
 only if
 <math display="block">