de Morgan's law

#1
I need help. I just can't figure out where to go with this one..

For two events A and B show that:

P(A intersection B) is greater than or equal to P(A) + P(B) - 1

Hint: Apply de Morgan's law and then the Bonferroni inequality.

(I don't know how to type the symbols out, so I used words instead)

Help is greatly appreciated... Extra 'bubbly' all around..
 

Mahi

New Member
#2
Hi
The De Morgan's law is:P(AUB) = P(A) + P(B) - P(A intersection B)

And we know that P(AUB) <= 1

Implies: P(AUB) = P(A) + P(B) - P(A intersection B) <= 1
Implies: P(A) + P(B) - P(A intersection B) <= 1
Implies: P(A) + P(B) - P(A intersection B) - 1 <= 1 - 1
Implies: P(A) + P(B) - P(A intersection B) - 1 <= 0
Implies: P(A) + P(B) - 1 <= P(A intersection B)
Thus,
P(A intersection B) >= P(A) + P(B) - 1.