Probability：練習問題

The events \(C\) and \(D\) are independent.
Given that \(\text{P}(C) = 0.4\) and \(\text{P}(D) = 0.7\), find \(\text{P}(C \cap D)\).

The events \(A\) and \(B\) are such that \(\text{P}(A) = 0.45\), \(\text{P}(B \mid A') = 0.20\), and \(\text{P}(A \cap B') = 0.30\).
Find \(\text{P}(B)\).

A bag contains 5 red balls and \(n\) blue balls. Two balls are drawn at random without replacement.
The probability that both balls are red is \(\frac{5}{33}\).
Find the value of \(n\) and the probability that the two balls are of different colors.

A bag contains 5 red balls, 3 blue balls, and 2 green balls. A ball is drawn at random from the bag.
Find the probability that the ball drawn is not red.

A bag contains 4 red beads and 6 green beads. Two beads are drawn at random one after another without replacement.
Find the probability that both beads are of the same colour.

Two events \( A \) and \( B \) are such that \( P(A) = 0.6 \) and \( P(B) = 0.2 \).

(a) State the value of \( P(A \cap B) \) if \( A \) and \( B \) are mutually exclusive.
(b) Find \( P(A \cup B) \) if \( A \) and \( B \) are mutually exclusive.
(c) Given instead that \( A \) and \( B \) are independent, find \( P(A \cup B) \).

A factory uses two machines, \( X \) and \( Y \), to manufacture light bulbs. Machine \( X \) produces \( 70\% \) of the bulbs and Machine \( Y \) produces the remaining \( 30\% \). It is known that \( 5\% \) of the bulbs produced by Machine \( X \) are defective and \( 8\% \) of the bulbs produced by Machine \( Y \) are defective.

A bulb is selected at random from the total production.

(a) Find the probability that the bulb is defective.
(b) Given that the selected bulb is defective, find the probability that it was produced by Machine \( X \). Give your answer as a fraction in its simplest form.