To solve the problem, we need to find [tex]\( P(A \cap B) \)[/tex] and [tex]\( P(A \cup B) \)[/tex] given that events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, with [tex]\( P(A) = 0.30 \)[/tex] and [tex]\( P(\bar{B}) = 0.40 \)[/tex].
### Step-by-Step Solution:
1. Find [tex]\( P(B) \)[/tex]:
Since [tex]\( P(\bar{B}) = 0.40 \)[/tex], we know that [tex]\( P(B) \)[/tex] is the complement of [tex]\( P(\bar{B}) \)[/tex].
[tex]\[
P(B) = 1 - P(\bar{B}) = 1 - 0.40 = 0.60
\][/tex]
2. Calculate [tex]\( P(A \cap B) \)[/tex]:
For independent events [tex]\( A \)[/tex] and [tex]\( B \)[/tex], the probability of their intersection, [tex]\( P(A \cap B) \)[/tex], is given by:
[tex]\[
P(A \cap B) = P(A) \times P(B)
\][/tex]
Substituting the given values:
[tex]\[
P(A \cap B) = 0.30 \times 0.60 = 0.18
\][/tex]
3. Calculate [tex]\( P(A \cup B) \)[/tex]:
The probability of the union of two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is given by:
[tex]\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\][/tex]
Substituting the known values:
[tex]\[
P(A \cup B) = 0.30 + 0.60 - 0.18 = 0.72
\][/tex]
Therefore, the solutions are:
(a) [tex]\( P(A \cap B) = 0.18 \)[/tex]
(b) [tex]\( P(A \cup B) = 0.72 \)[/tex]
