Lesson 1

The conditional probability of event \(A\) given event \(B\) is defined as follows: \[\Pr[A \mid B]=\frac{\Pr[A \cap B]}{\Pr[B]}\ .\] It is defined only if \(\Pr[B]>0\). Thus, when transitioning to conditional probability, we simply narrow the probability space to \(B\) (before narrowing, we could write the probability of any event \(C\) as \(\Pr[C \mid U]\), where \(U\) is the sample space). In other words, original probabilities represent fractions within the set \(U\); conditional probabilities represent fractions within the set \(B\), to which we have narrowed down. As shown in the figure below, under such narrowing, event probabilities can change in any way (both decrease and increase).

(a) The original sample space \(U\) is represented as a disjoint union of four events: \[U=\textcolor{#ff5349}{A}\sqcup \textcolor{#228B22}{B}\sqcup \textcolor{#DAA520}{C}\sqcup \textcolor{#4467F5}{D}.\] Accordingly, \[\Pr[\textcolor{#ff5349}{A}]=\frac{1}{10},\ \Pr[\textcolor{#228B22}{B}]=\frac{3}{10},\ \Pr[\textcolor{#DAA520}{C}]=\frac{4}{10},\ \Pr[\textcolor{#4467F5}{D}]=\frac{2}{10}.\] (b) Event \(E\) (to which the probability space is narrowed). Accordingly, \[\Pr[\textcolor{#ff5349}{A}\mid E]=0,\ \Pr[\textcolor{#228B22}{B}\mid E]=\frac{2}{7},\ \Pr[\textcolor{#DAA520}{C}\mid E]=\frac{4}{7},\ \Pr[\textcolor{#4467F5}{D}\mid E]=\frac{1}{7}.\] (c) The obtained conditional probabilities can conveniently be thought of simply as event probabilities within event \(E\).