Question

suppose that \\(a\\) and \\(b\\) are independent events such that \\(p(a) = 0.20\\) and \\(p(\overline{b}) = 0.40\\).

find \\(p(a \cap b)\\) and \\(p(a \cup b)\\). (if necessary, consult a list of formulas.)

(a) \\(p(a \cap b) = \\)
(b) \\(p(a \cup b) = \\)

Explanation:

Find the probability of event B

We are given the probability of the complement of \(B\), which is \(P(\overline{B}) = 0.40\). Using the concept of Probability of Events for complementary events:

$$ P(B) = 1 - P(\overline{B}) $$

$$ P(B) = 1 - 0.40 = 0.60 $$

Calculate the intersection probability

Since \(A\) and \(B\) are independent events, we apply the multiplication rule for Independent Events Probability:

$$ P(A \cap B) = P(A) \times P(B) $$

Substitute the known values \(P(A) = 0.20\) and \(P(B) = 0.60\):

$$ P(A \cap B) = 0.20 \times 0.60 = 0.12 $$

Calculate the union probability

To find the probability of the union of the two events, we use the general addition rule for Probability of Events:

$$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$

Substitute the calculated values into the formula:

$$ P(A \cup B) = 0.20 + 0.60 - 0.12 = 0.68 $$

Answer:

Question 1

(a) \(P(A \cap B) =\) <blank>0.12</blank>

Question 2

(b) \(P(A \cup B) =\) <blank>0.68</blank>