Independent, Mutually Exclusive, Combined and Successive Events：练习题

A bag contains \(3\) red tokens and \(7\) green tokens. A token is randomly selected, its colour is recorded, and it is then returned to the bag. A second token is then randomly selected. What is the probability that both tokens drawn are red?

A drawer contains \(6\) identical-looking batteries, but \(2\) of them are completely flat. If \(2\) batteries are selected at random without replacement, what is the probability that at least one of the selected batteries is flat?

In a competition, three archers \(A\), \(B\), and \(C\) shoot at a target simultaneously. Their probabilities of hitting the target are \(\frac{2}{3}\), \(\frac{3}{4}\), and \(\frac{4}{5}\) respectively. What is the probability that exactly two of the archers hit the target?

Two independent events, \(A\) and \(B\), have probabilities \(P(A) = 0.4\) and \(P(B) = 0.5\). What is the probability that both event \(A\) and event \(B\) occur, denoted as \(P(A \cap B)\)?

Students in a class are assessed in two subjects: Mathematics and Science. The probability that a randomly chosen student passes Mathematics is \(0.8\), and the probability that they pass Science is \(0.75\). If the passing of each subject is independent, what is the probability that a student passes exactly one of the two subjects?

Two fair six-sided dice are rolled simultaneously. What is the probability that the sum of the numbers shown on the two dice is a prime number? Express your answer as a simplified fraction.

Alice and Bob play a game where they take turns rolling a fair six-sided die. Alice goes first. Alice wins if she rolls a \(6\). Bob wins if he rolls an even number (\(2\), \(4\), or \(6\)). The game continues until one of them wins. Calculate the probability that Alice wins the game. Express your answer as a simplified fraction.

Three archers, A, B, and C, fire one arrow each at a target. The probabilities that they hit the target are \( 0.5 \), \( 0.6 \), and \( 0.7 \) respectively. What is the probability that the target is hit by exactly two archers? Express your answer as a decimal.

A box contains 7 red pens and 3 blue pens. Two pens are drawn at random from the box one by one without replacement.

(a) Find the probability that both pens drawn are red.
(b) Find the probability that the two pens drawn are of different colors.

A company manufactures light bulbs using two machines, \(M_1\) and \(M_2\). Machine \(M_1\) produces \(60\%\) of the total output and Machine \(M_2\) produces \(40\%\) of the total output. It is known that \(3\%\) of the light bulbs produced by \(M_1\) are defective, while \(5\%\) of the light bulbs produced by \(M_2\) are defective.

(a) A light bulb is selected at random from the total output. Find the probability that it is defective.
(b) Given that a randomly selected light bulb is defective, calculate the probability that it was produced by Machine \(M_1\).