Certainly! Let's solve the given problem step-by-step:
### Given:
- [tex]\( P(A) = 0.10 \)[/tex]
- [tex]\( P(\neg B) = 0.50 \)[/tex]
#### Step 1: Find [tex]\( P(B) \)[/tex]
Since [tex]\( \neg B \)[/tex] (not [tex]\( B \)[/tex]) is the complement of [tex]\( B \)[/tex], we can find [tex]\( P(B) \)[/tex] as follows:
[tex]\[ P(B) = 1 - P(\neg B) \][/tex]
Given [tex]\( P(\neg B) = 0.50 \)[/tex]:
[tex]\[ P(B) = 1 - 0.50 = 0.50 \][/tex]
#### Step 2: Find [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 the product of their individual probabilities:
[tex]\[ P(A \cap B) = P(A) \times P(B) \][/tex]
Given [tex]\( P(A) = 0.10 \)[/tex] and [tex]\( P(B) = 0.50 \)[/tex]:
[tex]\[ P(A \cap B) = 0.10 \times 0.50 = 0.05 \][/tex]
So, [tex]\( P(A \cap B) = 0.05 \)[/tex].
#### Step 3: Find [tex]\( P(A \cup B) \)[/tex]
For independent events [tex]\( A \)[/tex] and [tex]\( B \)[/tex], the probability of their union [tex]\( P(A \cup B) \)[/tex] is given by:
[tex]\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \][/tex]
Given [tex]\( P(A) = 0.10 \)[/tex], [tex]\( P(B) = 0.50 \)[/tex], and [tex]\( P(A \cap B) = 0.05 \)[/tex]:
[tex]\[ P(A \cup B) = 0.10 + 0.50 - 0.05 = 0.55 \][/tex]
So, [tex]\( P(A \cup B) = 0.55 \)[/tex].
### Final Answers:
(a) [tex]\( P(A \cap B) = 0.05 \)[/tex]
(b) [tex]\( P(A \cup B) = 0.55 \)[/tex]
