19M.2.SL.TZ2.S_9

pestleMathematicsAASLPaper 219M· sl-4-6-combined-mutually-exclusive-conditional-independence-prob-diagramssource ↗

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 $\sigma$ . 90 % of pilots fly less than 28 hours in a week.

Show that event A and event D are not independent.

[2]

Find ${\text{P}}\left( {A \cap D'} \right)$ .

[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 $\sigma$ .

[3]

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

[4]

Markscheme / solution

* 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 $0.595 \ne 0.65$ , $0.70 \times 0.85 \ne {\text{P}}\left( {A \cap D} \right)$

A and D are not independent AG N0

METHOD 2

calculation of ${\text{P}}\left( {D\left| A \right.} \right)$ (A1)

eg $\frac{{13}}{{14}}$ , 0.928

correct reasoning for their probabilities R1

eg $0.928 \ne 0.85$ , $\frac{{0.65}}{{0.7}}{\text{P}}\left( D \right)$

A and D are not independent AG N0

[2 marks]

correct working (A1)

eg ${\text{P}}\left( A \right) - {\text{P}}\left( {A \cap D} \right)$ , 0.7 − 0.65 , correct shading and/or value on Venn diagram

${\text{P}}\left( {A \cap D'} \right) = 0.05$ A1 N2

[2 marks]

b.i.

recognizing conditional probability (seen anywhere) (M1)

eg $\frac{{{\text{P}}\left( {D' \cap A} \right)}}{{{\text{P}}\left( A \right)}}$ , ${\text{P}}\left( {A\left| B \right.} \right)$

correct working (A1)

eg $\frac{{0.05}}{{0.7}}$

0.071428

${\text{P}}\left( {D'\left| A \right.} \right) = \frac{1}{{14}}$ , 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 $\sigma$ (A1)

eg $1.28155 = \frac{{28 - 25}}{\sigma }$ , $\frac{{28 - 25}}{{1.28155}}$

2.34091

$\sigma = 2.34$ A1 N2

[3 marks]

${\text{P}}\left( {X > 30} \right) = 0.0163429$ (A1)

valid approach (seen anywhere) (M1)

eg ${\left[ {{\text{P}}\left( {X > 30} \right)} \right]^2}$ , (0.01634)² , B(2, 0.0163429) , 2.67E-4 , 2.66E-4

0.0267090

0.0267 % A2 N3

[4 marks]

Examiners’ report

[N/A]

b.i.

[N/A]

b.ii.

[N/A]