4.2 Conditional Probability

Conditional Probability


Let

\[A:= \text{the event that Albert is giving the lecture}\] \[L:= \text{the event that Louis Reasoner goes to lecture}\]
where \[\Pr[A] = 0.8\] \[\Pr[L] = 0.4\]

Also, the probability that Louis goes to lecture, given that Albert is giving the lecture, is 0.3.

What is the probability that Albert is giving the lecture, given that Louis Reasoner goes to lecture?

Exercise 1

Straightforward use of the a posteriori probability formula. By definition, \(\Pr[A\;|\;L]\cdot\Pr[L] = \Pr[A \cap L] = \Pr[L \;|\; A]\cdot\Pr[A]\). Solving for \(\Pr[A\; |\; L]\) gives \(\Pr[A\; |\; L] = \frac{Pr[L\; |\; A]\cdot\Pr[A]}{Pr[L]} = \frac{0.3 \cdot 0.8}{0.4} = 0.6.\)