Solution: Application of the Law of Total Probability

(related to Problem: Broken Items in the Box)

We define the events

\(A:="\text{The first item drawn is not broken}"\)

\(B:="\text{The second item drawn is not broken}"\)

Because we have a Laplace experiment, the probabilities of \(A\) and \(\overline{A}\) are

\[p(A)=\frac {10-4}{10}=\frac 35;\quad\quad p(\overline{A})=1-p(A)=\frac 25.\]

Because the drawing is without replacement, the probabilities of \(B\) given \(A\) and \(\overline{A}\) are

\[p(B|A)=\frac {9-4}{9}=\frac 59;\quad\quad p(B|\overline{A})=\frac {9-3}{9}=\frac 69=\frac 23.\]

Note that the events \(A\) and \(\overline{A}\) are mutually exclusive and collectively exhaustive. It follows that we can apply the law of total probability and calculate the probability of \(B\):

\[p(B)=p(B|A)p(A)+ p(B|\overline{A})p(\overline{A})=\frac 56\cdot \frac 35+\frac 23\cdot \frac 25=\frac{27}{45}=\frac 35.\]

It turns out that the events \(A\) and \(B\) have the same probability.

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  1. Bosch, Karl: "Elementare Einf├╝hrung in die Wahrscheinlichkeitsrechnung", vieweg Studium, 1995, 6th Edition