Proof

(related to Proposition: Probability of Event Difference)

We first note that

\[A=(A\cap B)\cup (A\cap \overline B)=(A\cap B)\cup (A\setminus B).\]

Because the events \((A\cap B)\) and \((A\setminus B)\) are mutually exclusive, it follows from the definition of probability that

\[p(A)=p(A\cap B)+p(A\setminus B),\]

which gives us the required equation

\[p(A\setminus B)=p(A)-p(A\cap B).\]


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References

Bibliography

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