Each of the 180 students at a college plays exactly one of the piano, the guitar and the drums. The numbers of male and female students who play the piano, the guitar and the drums are given in the following table.

A student at the college is chosen at random.

(a)

Find the probability that the student plays the guitar.

[ 1 ]

(b)

Find the probability that the student is male given that the student plays the drums.

[ 2 ]

(c)

Determine whether the events 'the student plays the guitar' and 'the student is female' are independent, justifying your answer.

[ 2 ]

[Maximum number: 4]

Two ordinary fair dice, one red and the other blue, are thrown.
Event A is 'the score on the red die is divisible by 3 '.
Event B is 'the sum of the two scores is at least 9 '.

(a)

Find $\mathrm{P}(A \cap B)$ .

[ 2 ]

(b)

Hence determine whether or not the events A and B are independent.

[ 2 ]

[Maximum number: 1]

The score when two fair six-sided dice are thrown is the sum of the two numbers on the upper faces.

(a)

Show that the probability that the score is 4 is $\frac{1}{12}$ .

[ 1 ]

[Maximum number: 6]

Juan goes to college each day by any one of car or bus or walking. The probability that he goes by car is 0.2 , the probability that he goes by bus is 0.45 and the probability that he walks is 0.35 . When Juan goes by car, the probability that he arrives early is 0.6 . When he goes by bus, the probability that he arrives early is 0.1 . When he walks he always arrives early.

(a)

Draw a fully labelled tree diagram to represent this information.

[ 2 ]

(b)

Find the probability that Juan goes to college by car given that he arrives early.

[ 4 ]

[Maximum number: 5]

On any day, Kino travels to school by bus, by car or on foot with probabilities 0.2, 0.1 and 0.7 respectively. The probability that he is late when he travels by bus is x. The probability that he is late when he travels by car is 2 x and the probability that he is late when he travels on foot is 0.25 .

The probability that, on a randomly chosen day, Kino is late is 0.235 .

(a)

Find the value of x.

[ 3 ]

(b)

Find the probability that, on a randomly chosen day, Kino travels to school by car given that he is not late.

[ 2 ]

[Maximum number: 4]

A bag contains 9 blue marbles and 3 red marbles. One marble is chosen at random from the bag. If this marble is blue, it is replaced back into the bag. If this marble is red, it is not returned to the bag. A second marble is now chosen at random from the bag.

(a)

Find the probability that both the marbles chosen are red.

[ 1 ]

(b)

Find the probability that the first marble chosen is blue given that the second marble chosen is red.

[ 3 ]

[Maximum number: 5]

A total of 500 students were asked which one of four colleges they attended and whether they preferred soccer or hockey. The numbers of students in each category are shown in the following table.

(a)

Find the probability that a randomly chosen student is at Canton college and prefers hockey.

[ 1 ]

(b)

Find the probability that a randomly chosen student is at Devar college given that he prefers soccer.

[ 2 ]

(c)

One of the students is chosen at random. Determine whether the events 'the student prefers hockey' and 'the student is at Amos college or Benn college' are independent, justifying your answer.

[ 2 ]

[Maximum number: 2]

The time taken in minutes for a certain daily train journey has a normal distribution with standard deviation 5.8. For a random sample of 20 days the journey times were noted and the mean journey time was found to be 81.5 minutes.

(a)

Given that at least one of these intervals contains the population mean, find the probability that both intervals contain the population mean.

[ 2 ]

(a)

An arrangement of the 8 letters in the word RELEASED is chosen at random. Find the probability that the letters A and D are not together.

[ 4 ]

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