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Date May 2018 Marks available 2 Reference code 18M.2.AHL.TZ2.H_9
Level Additional Higher Level Paper Paper 2 Time zone Time zone 2
Command term Find Question number H_9 Adapted from N/A

Question

The number of taxis arriving at Cardiff Central railway station can be modelled by a Poisson distribution. During busy periods of the day, taxis arrive at a mean rate of 5.3 taxis every 10 minutes. Let T represent a random 10 minute busy period.

Find the probability that exactly 4 taxis arrive during T.

[2]
a.i.

Find the most likely number of taxis that would arrive during T.

[2]
a.ii.

Given that more than 5 taxis arrive during T, find the probability that exactly 7 taxis arrive during T.

[3]
a.iii.

During quiet periods of the day, taxis arrive at a mean rate of 1.3 taxis every 10 minutes.

Find the probability that during a period of 15 minutes, of which the first 10 minutes is busy and the next 5 minutes is quiet, that exactly 2 taxis arrive.

[6]
b.

Markscheme

X Po ( 5.3 )

P ( X = 4 ) = e 5.3 5.3 4 4 !      (M1)

= 0.164      A1

[2 marks]

a.i.

METHOD 1

listing probabilities (table or graph)      M1

mode X = 5 (with probability 0.174)     A1

Note: Award M0A0 for 5 (taxis) or mode = 5 with no justification.

 

METHOD 2

mode is the integer part of mean      R1

E(X) = 5.3 ⇒ mode = 5      A1

Note: Do not allow R0A1.

[2 marks]

a.ii.

attempt at conditional probability       (M1)

P ( X = 7 ) P ( X 6 ) or equivalent  ( = 0.1163 0.4365 )       A1

= 0.267       A1

[3 marks]

a.iii.

METHOD 1

the possible arrivals are (2,0), (1,1), (0,2)       (A1)

Y Po ( 0.65 )      A1

attempt to compute, using sum and product rule,      (M1)

0.070106… × 0.52204… + 0.026455… × 0.33932… + 0.0049916… × 0.11028…      (A1)(A1)

Note: Award A1 for one correct product and A1 for two other correct products.

= 0.0461       A1

[6 marks]

 

METHOD 2

recognising a sum of 2 independent Poisson variables eg Z = X + Y      R1

λ = 5.3 + 1.3 2

P(Z = 2) = 0.0461     (M1)A3

[6 marks]

 

b.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
a.iii.
[N/A]
b.

Syllabus sections

Topic 4—Statistics and probability » SL 4.8—Binomial distribution
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Topic 4—Statistics and probability

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