exercise:Fa9dae7050: Difference between revisions
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(Created page with "<div class="d-none"><math> \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> Prove that for any three events <math>A</math>, <math>B</math>, <math>C</math>, each having positive probability, and with the property that <math>P(A \cap B) > 0</math>, <math display="block"> P(A \cap B \cap C) = P(A)P(B|A)P(C|A \cap B)\ . </...") |
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\newcommand{\secstoprocess}{\all} | \newcommand{\secstoprocess}{\all} | ||
\newcommand{\NA}{{\rm NA}} | \newcommand{\NA}{{\rm NA}} | ||
\newcommand{\mathds}{\mathbb}</math></div> Prove that for any three events <math>A</math>, <math>B</math>, <math>C</math>, each having positive | \newcommand{\mathds}{\mathbb}</math></div> Prove that for any three events <math>A</math>, <math>B</math>, <math>C</math>, each having positive probability, and with the property that <math>P(A \cap B) > 0</math>, | ||
probability, and with the property that <math>P(A \cap B) > 0</math>, | |||
<math display="block"> | <math display="block"> | ||
P(A \cap B \cap C) = P(A)P(B|A)P(C|A \cap B)\ . | P(A \cap B \cap C) = P(A)P(B|A)P(C|A \cap B)\ . | ||
</math> | </math> | ||
Latest revision as of 23:45, 12 June 2024
[math]
\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]
Prove that for any three events [math]A[/math], [math]B[/math], [math]C[/math], each having positive probability, and with the property that [math]P(A \cap B) \gt 0[/math],
[[math]]
P(A \cap B \cap C) = P(A)P(B|A)P(C|A \cap B)\ .
[[/math]]