Question

suppose that \\(a\\) and \\(b\\) are independent events such that \\(p(a) = 0.50\\) and \\(p(\overline{b}) = 0.60\\).

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

Using the Probability of Events knowledge point

$$ P(B) = 1 - P(\overline{B}) = 1 - 0.60 = 0.40 $$

Calculate the intersection probability

Using the Independent Events Probability knowledge point

$$ P(A \cap B) = P(A) \times P(B) = 0.50 \times 0.40 = 0.20 $$

Calculate the union probability

Using the Probability of Events knowledge point

$$ P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.50 + 0.40 - 0.20 = 0.70 $$

Answer:

Suppose that \(A\) and \(B\) are independent events such that \(P(A) = 0.50\) and \(P(\overline{B}) = 0.60\).
Find \(P(A \cap B)\) and \(P(A \cup B)\).

(a) \(P(A \cap B) =\) <blank>0.20</blank>
(b) \(P(A \cup B) =\) <blank>0.70</blank>