Final answer:
To find P(a ∪ b) when a and b are independent, use the formula P(a ∪ b) = P(a) + P(b) - P(a ∩ b). To find P(a | b) when a and b are mutually exclusive, use the formula P(a | b) = P(a ∩ b) / P(b).
Step-by-step explanation:
(a) To find P(a ∪ b) when a and b are independent, we need to use the formula P(a ∪ b) = P(a) + P(b) - P(a ∩ b). Since a and b are independent, P(a ∩ b) = P(a)P(b). Therefore, P(a ∪ b) = P(a) + P(b) - P(a)P(b). Substituting the given values, P(a) = 0.3 and P(b) = 0.6, we get P(a ∪ b) = 0.3 + 0.6 - (0.3)(0.6) = 0.3 + 0.6 - 0.18 = 0.72.
(b) To find P(a | b) when a and b are mutually exclusive, we use the formula P(a | b) = P(a ∩ b) / P(b). Since a and b are mutually exclusive, P(a ∩ b) = 0. Therefore, P(a | b) = 0 / P(b) = 0.