Homework SourseSuppose that PA 34 and PB 13 a Show that 112 is less than
Suppose that P(A) = 3/4 and P(B) = 1/3.
(a) Show that 1/12 is less than or equal to P(A intersection B) is less than or equal to 1/3
(b) Find the minimum and maximum of P(A union B)
Solution
A)
As
0< P(A U B) <1
Then
0 < P(A) + P(B) - P(A n B) < 1
0 < 3/4 + 1/3 - P(A n B) < 1
0 < 13/12 - P(A n B) < 1
1/12 < P(A n B) < 13/12
However, note that
P(A n B) < P(B) = 1/3
Thus,
1/12 < P(A n B) < 1/3 [ANSWER]
B)
As
P(A U B) = P(A) + P(B) - P(A n B)
and
1/12 < P(A n B) < 1/3
Then
3/4 + 1/3 - P(A n B)max < P(A U B) < 3/4 + 1/3 - P(A n B)min
3/4 + 1/3 - 1/3 < P(A U B) < 3/4 + 1/3 - 1/12
3/4 < P(A U B) < 1 [ANSWER, minimum of 3/4, maximum of 1]
