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A-Level CAIE Mathematics AS5.3 ProbabilityQuestion Bank

[Maximum number: 5]

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.

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 P(A∩B)\mathrm{P}(A \cap B).

[ 2 ]
(b)

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

[ 2 ]
[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: 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 112\frac{1}{12}.

[ 1 ]
[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.

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 ]
(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 ]
[Maximum number: 5]

The probability that a student at a large music college plays in the band is 0.6 . For a student who plays in the band, the probability that she also sings in the choir is 0.3 . For a student who does not play in the band, the probability that she sings in the choir is x. The probability that a randomly chosen student from the college does not sing in the choir is 0.58 .

(a)

Find the value of x.

Two students from the college are chosen at random.

[ 3 ]
(b)

Find the probability that both students play in the band and both sing in the choir.

[ 2 ]
[Maximum number: 2]

A bag contains 5 red balls and 3 blue balls. Sadie takes 3 balls at random from the bag, without replacement. The random variable X represents the number of red balls that she takes.

(a)

Show that the probability that Sadie takes exactly 1 red ball is 1556\frac{15}{56}.

[ 2 ]
[Maximum number: 2]

A book club sends 6 paperback and 2 hardback books to Mrs Hunt. She chooses 4 of these books at random to take with her on holiday. The random variable X represents the number of paperback books she chooses.

(a)

Show that the probability that she chooses exactly 2 paperback books is 314\frac{3}{14}.

[ 2 ]
[Maximum number: 6]

On each day that Alexa goes to work, the probabilities that she travels by bus, by train or by car are 0.4,0.35 and 0.25 respectively. When she travels by bus, the probability that she arrives late is 0.55 . When she travels by train, the probability that she arrives late is 0.7 . When she travels by car, the probability that she arrives late is x.

On a randomly chosen day when Alexa goes to work, the probability that she does not arrive late is 0.48 .

(a)

Find the value of x.

[ 3 ]
(b)

Find the probability that Alexa travels to work by train given that she arrives late.

[ 3 ]
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