# 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.

Github: ### References

#### Bibliography

1. Bosch, Karl: "Elementare Einführung in die Wahrscheinlichkeitsrechnung", vieweg Studium, 1995, 6th Edition