3.4 Conditional probability

Conditional probability is the probability of event \(A\) after observing the occurrence of an event \(B\). The probability of \(A\) given \(B\) is

\[\begin{equation} Pr(A|B) = \dfrac{Pr(A \cap B)}{Pr(B)}, \; \; Pr(B) \ne 0 \tag{3.32} \end{equation}\]

Analogously \[\begin{equation} Pr(B|A) = \dfrac{Pr(A \cap B)}{Pr(A)}, \; \; Pr(A) \ne 0 \tag{3.33} \end{equation}\]

Example 3.19 (Conditional probability) A balanced die is rolled, and we wish to observe the event \(A\): ‘side 2’. The person who threw the dice also gives information \(B\): ‘the side is even’. Therefore, \[ Pr(B) = \frac{1}{2}, \] \[ Pr(A \cap B) = \frac{1}{6}, \] \[ Pr(A|B) = \dfrac{1/6}{1/2} = \dfrac{1}{3}, \] \[ Pr(A^{c}|B) = 1 - \dfrac{1}{3} = \dfrac{2}{3}. \]

The Equations (3.32) and (3.33) result in the chain rule , or the probability of the intersection event:

\[\begin{equation} Pr(A \cap B) = Pr(A) \cdot Pr(B|A) = Pr(B) \cdot Pr(A|B) \tag{3.34} \end{equation}\]

Three events \[\begin{equation} Pr(A \cap B \cap C) = Pr(A) \cdot Pr(B|A) \cdot Pr(C|A \cap B) \tag{3.35} \end{equation}\]

General form (Pfeiffer and Schum 1973, 90) \[\begin{equation} Pr(\cap_{i=1}^k A_i) = Pr(A_1) \cdot Pr(A_2|A_1) \cdot Pr(A_3|A_1 \cap A_2) \cdots Pr(A_k|A_1 \cap \cdots \cap A_{k-1}) \tag{3.36} \end{equation}\]

Exercise 3.8 Redo Example 3.19 considering the information \(C\): ‘the face is odd’. Calculate:

  1. \(Pr(C)\)
  2. \(Pr(A \cap C)\)
  3. \(Pr(A \mid C)\)
  4. \(Pr(A^C \mid C)\)


3.4.1 Independence

When \[\begin{equation} Pr(A|B) = \dfrac{Pr(A) \cdot Pr(B)}{Pr(B)} = Pr(A) \tag{3.37} \end{equation}\] \(A\) and \(B\) are said to be independent, symbolized by \(A \perp\!\!\!\perp B\). This indicates that observing \(B\) does not change the probability with respect to \(A\).

The probability properties still hold, allowing us to do, for example, \[\begin{equation} Pr(A|B) = 1 - Pr(A^{c}|B) \tag{3.38} \end{equation}\]

3.4.2 Conditional independence

Conditional Independence or Death! based on Independence or Death! by Pedro Americo (1888)
Conditional Independence or Death! based on Independence or Death! by Pedro Americo (1888)

\(\\\) Let \(A\), \(B\) and \(C\) be events of \(\Omega\). \(A\) and \(B\) are conditionally independent given \(C\) if and only if

\[\begin{equation} Pr(A|B,C) = Pr(A|C) \tag{3.39} \end{equation}\]

For more details see (Basu and Pereira 2011) and (Studený 2005). For a cultural little moment see Paulo César Garcez Marins | Provoca | 06/06/2023.


Basu, D, and Carlos AB Pereira. 2011. “Conditional Independence in Statistics.” Selected Works of Debabrata Basu, 371–84.
Pfeiffer, Paul E, and David A Schum. 1973. Introduction to Applied Probability. Elsevier.
Studený, Milan. 2005. Probabilistic Conditional Independence Structures. Springer Science & Business Media.