Page 26 - Probability, Random Variables and Random Processes

PROBABILITY [CHAP 1

1.25. For any three events A,, A,, and A,, show that
P(Al u A, u A,) = P(Al) + P(A,) + P(A,) - P(A, n A,)
- P(Al n A,) - P(A, n A,) + P(Al n A, n A,)

Let B = A, u A,. By Eq. (1.29), we have

Using distributive law (1.1 2), we have
A, n B = A, n (A, u A,) = (A, n A,) u (A, n A,)
Applying Eq. (1.29) to the above event, we obtain
P(Al n B) = P(Al n A,) + P(Al n A,) - P[(Al n A,) n (A, n A,)]
="P(Al n A,) + P(Al n A,) - P(Al n A, n A,)
Applying Eq. (1.29) to the set B = A, u A,, we have
P(B) = P(A, u A,) = P(A,) + P(A,) - P(A, n A,)
Substituting Eqs. (1.69) and (1.68) into Eq. (1.67), we get
P(Al u A, u A,) = P(Al) + P(A,) + P(A,) - P(A, n A,) - P(A, n A,)
- P(A, n A,) + P(Al n A, n A,)

1.26. Prove that

which is known as Boole's inequality.
We will prove Eq. (1 .TO) by induction. Suppose Eq. (1.70) is true for n = k.

Then

Thus Eq. (1.70) is also true for n = k + 1. By Eq. (1.33), Eq. (1.70) is true for n = 2. Thus, Eq. (1.70) is true
for n 2 2.

1.27. Verify Eq. (1.31).
Again we prove it by induction. Suppose Eq. (1.31) is true for n = k.

Then

Using the distributive law (1.1 6), we have