Show that for the case of pairwise mutually exclusive if it

Show that for the case of pairwise mutually exclusive if it is true that:

Let A1, A2 ....be a pairwise disjoint events.

i.e. no common element willbe there between Ai and Aj when i not equals j.

Consider P(A1UA2) = P(A1)+P(A2)-P(A1A2)

But P(A1A2) =0

Hence P(A1UA2) = P(A1)+P(A2)

Hence the given stt is true for n =2.

Let us prove by induction.

Let P(k) be true.

i.e. P(A1UA2...UAk) = P(A1)+P(A2)+... P(Ak-1)+P(Ak)

Consider for k+1

P(A1UA2...UAkUAk+1)= P(A1UA2...UAk)+P(Ak+1)-P(A1UA2...UAk intersection Ak+1)

= P(A1)+P(A2)+... P(Ak-1)+P(Ak)+P(Ak+1) +0 as pairwise disjoint.

Hence true for k+1

Thus by PMI proved for all natural numbers.