Sure, let's solve this step by step:
Given:
- [tex]\( P(\bar{A}) = 0.10 \)[/tex] (the probability that event [tex]\(A\)[/tex] does not occur)
- [tex]\( P(B) = 0.60 \)[/tex] (the probability that event [tex]\(B\)[/tex] occurs)
Step 1: Find [tex]\( P(A) \)[/tex]
Since [tex]\( P(\bar{A}) \)[/tex] is the probability that event [tex]\( A \)[/tex] does not occur, we can find [tex]\( P(A) \)[/tex] by subtracting [tex]\( P(\bar{A}) \)[/tex] from 1.
[tex]\[ P(A) = 1 - P(\bar{A}) \][/tex]
Substitute the given value:
[tex]\[ P(A) = 1 - 0.10 = 0.90 \][/tex]
Step 2: Find [tex]\( P(A \cap B) \)[/tex]
Since [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent events, the probability of both [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occurring, denoted [tex]\( P(A \cap B) \)[/tex], is the product of their individual probabilities:
[tex]\[ P(A \cap B) = P(A) \times P(B) \][/tex]
Substitute the values:
[tex]\[ P(A \cap B) = 0.90 \times 0.60 = 0.54 \][/tex]
Step 3: Find [tex]\( P(A \cup B) \)[/tex]
The probability of either [tex]\( A \)[/tex] or [tex]\( B \)[/tex] occurring, denoted [tex]\( P(A \cup B) \)[/tex], can be found using the formula:
[tex]\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \][/tex]
Substitute the values:
[tex]\[ P(A \cup B) = 0.90 + 0.60 - 0.54 = 0.96 \][/tex]
Final Answers:
(a) The probability of both [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occurring (the intersection) is:
[tex]\[ P(A \cap B) = 0.54 \][/tex]
(b) The probability of either [tex]\( A \)[/tex] or [tex]\( B \)[/tex] occurring (the union) is:
[tex]\[ P(A \cup B) = 0.96 \][/tex]
