Probability of combined events：練習問題

In a tree diagram representing two rounds of a game, a node represents the possible outcomes of the first round. If the branch for Win has a probability of \( \frac{2}{7} \), what is the probability that must be written on the branch representing Lose originating from the same node?

A fair coin is tossed and a fair six-sided die is rolled. What is the probability of getting a Head on the coin and an even number on the die?

Two independent events \(A\) and \(B\) have probabilities \(P(A)\) and \(P(B)\). Given that \(P(A \cap B) = 0.12\) and \(P(A \cup B) = 0.58\), and knowing that \(P(A) > P(B)\), find the value of \(P(A)\).

The probability that it will rain tomorrow is \( 0.3 \). The probability that a student will be late for school is \( 0.2 \). Assuming these events are independent, find the probability that it rains and the student is late.

Two fair four-sided dice, each numbered 1 to 4, are rolled. Using a sample space diagram or otherwise, find the probability that the sum of the numbers on the two dice is exactly 5.

In a group of 20 students, 12 study History, 10 study Geography, and 4 study both subjects. If a student is chosen at random, find the probability that they study at least one of these two subjects.

A spinner has sections coloured red, blue, or green. The probability of landing on red is 0.35 and the probability of landing on blue is 0.4. If landing on red, blue, or green are the only possible outcomes, calculate the probability of the spinner landing on green.

A bag contains 5 red balls and 3 blue balls. Two balls are chosen at random without replacement. Calculate the probability that both balls are blue.

A bag contains 6 black pens and 4 blue pens. A pen is taken at random from the bag, its colour is noted, and it is returned to the bag. A second pen is then taken at random.

(a) Write down the probability that the first pen is black.

(b) Calculate the probability that both pens are black.

(c) Calculate the probability that exactly one pen is black and one pen is blue.

A bag contains 8 red, 5 blue, and 7 green counters. A counter is chosen at random from the bag.

(a) Find the probability of choosing a red counter.

(b) Find the probability of not choosing a blue counter.

(c) If a counter is chosen, its colour noted, and then replaced. Then another counter is chosen at random. What is the probability of choosing two counters of different colours?