Given that:
S = {e1, e2, e3, e4, e5, e6, e7, e8, e9}
A = {e1, e5, e8} and B = {e2, e5, e8, e9}
P(e1) = P(e2) = .08, P(e3) = P(e4) = P(e5) = .1
P(e6) = P(e7) = .2, P(e8) = P(e9) = .07
(a) To find: P(A), P(B) and P(A ⋂ B)
A = {e1, e5, e8}
P(A) = P(e1) + P(e5) + P(e8)
⇒ P(A) = 0.08 + 0.1 + 0.07 [given]
⇒ P(A) = 0.25
B = {e2, e5, e8, e9}
P(B) = P(e2) + P(e5) + P(e8) + P(e9)
⇒ P(B) = 0.08 + 0.1 + 0.07 + 0.07 [given]
⇒ P(B) = 0.32
Now, we have to find P(A ⋂ B)
A = {e1, e5, e8} and B = {e2, e5, e8, e9}
∴ A ⋂ B = {e5, e8}
⇒ P(A ⋂ B) = P(e5) + P(e8)
= 0.1 + 0.07
= 0.17
(b) To find: P(A ⋃ B)
By General Addition Rule:
P(A ⋃ B) = P(A) + P(B) – P(A ⋂ B)
from part (a), we have
P(A) = 0.25, P(B) = 0.32 and P(A ⋂ B) = 0.17
Putting the values, we get
P(A ⋃ B) = 0.25 + 0.32 – 0.17
= 0.40
(c) A = {e1, e5, e8} and B = {e2, e5, e8, e9}
∴ A ⋃ B = {e1, e2, e5, e8, e9}
⇒ P(A ⋃ B) = P(e1) + P(e2) + P(e5) + P(e8) + P(e9)
= 0.08 +0.08 + 0.1 + 0.07 + 0.07
= 0.40
= 0.08 + 0.1 + 0.1 + 0.2 + 0.2 [given]
= 0.68
