Conditional probability：練習問題

In a class of 30 students, 18 study Mathematics, 15 study Science, and 7 study both. A student is chosen at random from the class. What is the probability that the student studies Mathematics, given that they study Science?

A bag contains 4 red and 6 blue marbles. Two marbles are drawn at random without replacement. What is the probability that both marbles are blue, given that at least one of them is blue?

A survey indicates that 60% of people own a smartphone (S) and 30% own a tablet (T). It is also found that 20% of people own both a smartphone and a tablet. What is the probability that a randomly chosen person owns a tablet, given that they own a smartphone?

Three machines A, B, and C produce 20%, 30%, and 50% of the total output, respectively. The percentages of defective items produced by these machines are 2%, 3%, and 1% respectively. If a randomly selected item is found to be defective, what is the probability that it was produced by machine B?

In a sports club of 50 members, 30 play tennis, 20 play badminton, and 10 play both sports. A member is chosen at random. What is the probability that the member plays badminton, given that they play tennis?

In a group of 50 students, 30 study Mathematics, 20 study Physics, and 10 study both. A student is chosen at random from those who study Mathematics. What is the probability that this student also studies Physics?

In a school, 60% of students own a smartphone. Of those who own a smartphone, 80% also own a tablet. Of those who do not own a smartphone, 20% own a tablet. Find the probability that a randomly selected student owns a smartphone, given that they own a tablet.

In a group of athletes, \( 40\% \) use a specific supplement. The probability of winning a race is \( 0.7 \) for those using the supplement and \( 0.3 \) for those who do not. If an athlete is chosen at random and they won their last race, calculate the probability that they used the supplement.

A box contains 10 chocolates: 6 with soft centres and 4 with hard centres. Two chocolates are chosen at random, one after the other, without replacement.

(a) Draw a tree diagram to represent all possible outcomes and their probabilities.

(b) Find the probability that both chocolates have soft centres.

(c) Find the probability that one chocolate has a soft centre and the other has a hard centre.

(d) Given that the second chocolate chosen has a hard centre, find the probability that the first chocolate chosen had a soft centre.

A survey was conducted among 120 people about their preference for coffee or tea, categorised by gender. The results are shown in the table below:

A person is chosen at random from the surveyed group.

(a) Find the probability that the person prefers coffee.

(b) Find the probability that the person is female and prefers tea.

(c) Given that the person chosen prefers tea, find the probability that the person is male.

(d) Are the events 'being male' and 'preferring coffee' independent? Justify your answer with calculations.