More about probability：練習問題

A card is drawn at random from eight cards numbered 1, 2, 3, 4, 5, 6, 7 and 8. Let \(A\) be the event that the number drawn is a prime number and \(B\) be the event that the number drawn is greater than 5. Find \(P(A \cup B)\).

For two events \(A\) and \(B\), it is given that \(P(A) = 0.7\), \(P(B) = 0.4\) and \(P(A \cup B) = 0.8\). Find \(P(A | B)\).

Bag A contains 3 red balls and 2 blue balls. Bag B contains 4 red balls and 3 blue balls. A fair coin is tossed. If it shows heads, two balls are drawn one after another without replacement from Bag A. If it shows tails, two balls are drawn one after another without replacement from Bag B. What is the probability that both balls drawn are red?

Six students, including John and Mary, are randomly arranged in a row for a photo. Find the probability that John and Mary are standing next to each other.

Let A and B be two events such that \(P(A) = 0.4\), \(P(B) = 0.6\), and \(P(A \cup B) = 0.8\). Find \(P(A' | B)\).

A fair six-sided die is rolled. What is the probability of not rolling a prime number?

A bag contains 5 red balls and 3 blue balls. If 3 balls are drawn at random without replacement, what is the probability that at least one ball is blue?

Bag A contains 5 red balls and 3 blue balls. Bag B contains 4 red balls and 6 blue balls. A fair die is rolled. If the outcome is a prime number (2, 3, 5), a ball is drawn randomly from Bag A and transferred to Bag B. If the outcome is not a prime number, a ball is drawn randomly from Bag B and transferred to Bag A. After the transfer, a ball is drawn from the bag that received a ball. Given that this final ball drawn is red, what is the probability that the initial die roll was a prime number?

A box contains 4 gold medals, 3 silver medals and 3 bronze medals. Three medals are randomly drawn from the box one by one with replacement.

(a) Find the probability that all three medals drawn are gold.
(b) Find the probability that at least one gold medal is drawn.
(c) Suppose the three medals are drawn without replacement. Find the probability that all three medals drawn are gold.
(d) Comparing the results in (a) and (c), which drawing method has a higher probability of getting three gold medals?

In a school, $$70\%$$ of students participate in sports ($$S$$) and $$40\%$$ participate in music ($$M$$). It is known that $$15\%$$ of students participate in neither sports nor music.

(a) Find the probability that a randomly selected student participates in both sports and music.

(b) Find the probability that a student participates in music given that they participate in sports.

(c) Are the events 'participating in sports' and 'participating in music' independent? Justify your answer.