Exam-style Questions: Probability Distributions

1. Two fair dice are thrown. If the scores are unequal, the larger of the two scores is recorded. If the scores are equal then that score is recorded. Let X denote the number recorded.

a) show that P(X = 2) = 1/12 and draw up a table showing the probability distribution of X

(4 marks)

b) find the mean and variance of

(4 marks)

(Marks available: 8)

Answer

Answer outline and marking scheme for question: 1

a) P(X = 2) = P( 1 and 2) or P(2 and 1) or P(2 and 2)

= (1/6 x 1/6) + (1/6 x 1/6) + (1/6 x 1/6) = 3/36 A1

= 1/12 as required A1

x 1 2 3 4 5 6
P(X=x) 1/36 1/12 5/36 7/36 9/36 11/36

(4 marks)

b) E(X) = (1 x 1/36) + (2 x 1/12) + (3 x 5/36) + (4 x 7/36) + (5 x 9/36) + (6 x 11/36)

= 161/36

= 4.47 (2dp)

for Var(X) we first need

E(X2) = (12 x 1/36) + (22 x 1/12) + (32 x 5/36) + (42 x 7/36) + (52 x 9/36) + (62 x 11/36)

= 791/36 = 21.97 (2dp)

Var(X) = E(X2) - E(X)2

= 21.97 - 4.472

= 1.97

(4 marks)

(Marks available: 8)

2. It is known that 65% of sixth formers think their maths teacher is cool. In a certain sample, 10 pupils are picked at random and it is required to calculate the probability that less than 3 of the 10 think that their teacher is cool. Name a probability distribution that can be used for modelling this situation stating one necessary assumption for this model to be valid.

a) use the model to calculate the required probability

(4 marks)

b) calculate the mean and variance of this distribution

(3 marks)

(Marks available: 7)

Answer

Answer outline and marking scheme for question: 2

Binomial distribution with n = 10 and probability of success, p = 0.65
assume each pupil picked is independent of each other

a) using X ~ B(10, 0.65)

need P(X

=0.3510 + 10(0.35)9(0.65)10C2 + (0.35)8(0.65)2

= 0.0048 (2sf)

(4 marks)

b) E(X) = n x p = 10 x 0.65 = 6.5 A1

Var(X) = n x p x q = 10 x 0.65 x 0.35 = 2.275

(3 marks)

(Marks available: 7)

3. When I ask a sixth form class a question the probability that I get an answer is 0.36. if I don't get an answer first time I keep trying until I am blessed with an answer. Let X denote the number of attempts I have to make in order to get an answer. Stating any assumption, identify the probability distribution of X.

(2 marks)

hence calculate:

a) P(X = 5)

(2 marks)

b) P(X = 4)

(3 marks)

c) E(X) and Var(X)

(3 marks)

(Marks available: 10)

Answer

Answer outline and marking scheme for question: 3

3. Geometric distribution with probability of success, p = 0.36
assume independence of sixth formers asked

(2 marks)

a) P(X = 5) = (0.64)4 x 0.36 = 0.0604 (3sf)

(2 marks)

b) X ~ Geo(0.36)

P(X = 4) = P(X > 5)

= q5 = 0.645

= 0.107 (3sf)

(3 marks)

c) E(X) = 1/p = 1/0.36 = 2.78 (3sf)

Var(X) = q/p2

= 0.64/0.362 = 4.94

(3 marks)

(Marks available: 10)