Further probability：练习题

For two events \(A\) and \(B\), it is given that \(P(A) = 0.3\), \(P(B) = 0.5\), and \(P(A | B) = 0.4\). Calculate the conditional probability \(P(B | A)\).

The conditional probabilities for event \(A\) given event \(B\) and its complement \(B'\) are \(P(A | B) = 0.7\) and \(P(A | B') = 0.4\). If \(P(B) = 0.3\), calculate the value of \(P(B | A)\).

Two events \(A\) and \(B\) are mutually exclusive. Given that \(P(A) = 0.25\) and \(P(B) = 0.45\), find the value of \(P(A' \cap B')\).

For two events \(A\) and \(B\), it is given that \(P(A | B) = 0.5\), \(P(B | A) = 0.4\), and \(P(A \cap B) = 0.2\). Calculate \(P(A' \cap B')\).

Events \(A\) and \(B\) are independent such that \(P(A) = 0.6\) and \(P(B) = 0.5\). Calculate \(P(A \cup B)\).

Events \(A\) and \(B\) are independent such that \(P(A) = 0.4\) and \(P(A \cup B) = 0.7\). Calculate the probability \(P(B)\).

Given that \(P(A) = 0.6\), \(P(B) = 0.5\), and \(P(A | B) = 0.8\), find the conditional probability \(P(B | A')\).

Three events \(A\), \(B\), and \(C\) are defined. Events \(A\) and \(B\) are mutually exclusive with \(P(A) = 0.3\) and \(P(B) = 0.4\). If event \(C\) is independent of the event \((A \cup B)\) and \(P(C) = 0.5\), find \(P((A \cup B) \cap C)\).

For two events \(A\) and \(B\), it is given that \(P(A) = 0.35\), \(P(B) = 0.45\), and \(P(A \cup B) = 0.6\).

(a) Calculate \(P(A \cap B)\).
(b) Find the conditional probability \(P(A | B)\).
(c) Determine, with a reason, whether events \(A\) and \(B\) are independent.

In a factory, machine A produces 60% of the total output and machine B produces the remaining 40%. The probability of a defective component being produced by machine A is 0.02, while the probability for machine B is 0.05.

(a) A component is selected at random from the total output. Find the probability that it is defective.
(b) Given that a randomly selected component is found to be defective, find the probability that it was produced by machine A.
(c) Two components are selected at random. Find the probability that exactly one is defective.