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Date May 2019 Marks available 3 Reference code 19M.2.SL.TZ2.S_9
Level Standard Level Paper Paper 2 Time zone Time zone 2
Command term Find Question number S_9 Adapted from N/A

Question

At Penna Airport the probability, P(A), that all passengers arrive on time for a flight is 0.70. The probability, P(D), that a flight departs on time is 0.85. The probability that all passengers arrive on time for a flight and it departs on time is 0.65.

The number of hours that pilots fly per week is normally distributed with a mean of 25 hours and a standard deviation σ . 90 % of pilots fly less than 28 hours in a week.

Show that event A and event D are not independent.

[2]
a.

Find P ( A D ) .

[2]
b.i.

 Given that all passengers for a flight arrive on time, find the probability that the flight does not depart on time.

[3]
b.ii.

Find the value of σ .

[3]
c.

All flights have two pilots. Find the percentage of flights where both pilots flew more than 30 hours last week.

[4]
d.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

METHOD 1

multiplication of P(A) and P(D)     (A1)

eg   0.70 × 0.85,  0.595

correct reasoning for their probabilities       R1

eg    0.595 0.65 ,    0.70 × 0.85 P ( A D )

A and D are not independent      AG N0

 

METHOD 2

calculation of  P ( D | A )        (A1)

eg    13 14 ,  0.928

correct reasoning for their probabilities       R1

eg    0.928 0.85 ,    0.65 0.7 P ( D )

A and D are not independent      AG N0

[2 marks]

a.

correct working       (A1)

eg   P ( A ) P ( A D ) ,  0.7 − 0.65 , correct shading and/or value on Venn diagram

P ( A D ) = 0.05        A1  N2

[2 marks]

 

b.i.

recognizing conditional probability (seen anywhere)       (M1)

eg    P ( D A ) P ( A ) ,   P ( A | B )

correct working       (A1)

eg     0.05 0.7

0.071428

P ( D | A ) = 1 14 , 0.0714     A1  N2

[3 marks]

b.ii.

finding standardized value for 28 hours (seen anywhere)       (A1)

eg    z = 1.28155

correct working to find σ        (A1)

eg     1.28155 = 28 25 σ 28 25 1.28155

2.34091

σ = 2.34      A1  N2

[3 marks]

c.

P ( X > 30 ) = 0.0163429        (A1)

valid approach (seen anywhere)        (M1)

eg    [ P ( X > 30 ) ] 2 ,  (0.01634)2 ,  B(2, 0.0163429) , 2.67E-4 , 2.66E-4

0.0267090

0.0267 %    A2  N3

[4 marks]

d.

Examiners report

[N/A]
a.
[N/A]
b.i.
[N/A]
b.ii.
[N/A]
c.
[N/A]
d.

Syllabus sections

Topic 4—Statistics and probability » SL 4.6—Combined, mutually exclusive, conditional, independence, prob diagrams
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Topic 4—Statistics and probability » SL 4.9—Normal distribution and calculations
Topic 4—Statistics and probability

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