19N.2.SL.TZ0.S_9

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

SpeedWay airline flies from city ${\text{A}}$ to city ${\text{B}}$ . The flight time is normally distributed with a mean of $260$ minutes and a standard deviation of $15$ minutes.

A flight is considered late if it takes longer than $275$ minutes.

The flight is considered to be on time if it takes between $m$ and $275$ minutes. The probability that a flight is on time is $0.830$ .

During a week, SpeedWay has $12$ flights from city ${\text{A}}$ to city ${\text{B}}$ . The time taken for any flight is independent of the time taken by any other flight.

Calculate the probability a flight is not late.

[2]

Find the value of $m$ .

[3]

Calculate the probability that at least $7$ of these flights are on time.

[3]

c.i.

Given that at least $7$ of these flights are on time, find the probability that exactly $10$ flights are on time.

[4]

c.ii.

SpeedWay increases the number of flights from city ${\text{A}}$ to city ${\text{B}}$ to $20$ flights each week, and improves their efficiency so that more flights are on time. The probability that at least $19$ flights are on time is $0.788$ .

A flight is chosen at random. Calculate the probability that it is on time.

[3]

Markscheme / solution

valid approach (M1)

eg ${\text{P}}\left( {X < 275} \right)$ , $1 - 0.158655$

$0.841344$

$0.841$ A1 N2

[2 marks]

valid approach (M1)

eg ${\text{P}}\left( {X < 275} \right) - {\text{P}}\left( {X < m} \right) = 0.830$

correct working (A1)

eg ${\text{P}}\left( {X < m} \right) = 0.0113447$

$225.820$

$226$ (minutes) A1 N3

[3 marks]

evidence of recognizing binomial distribution (seen anywhere) (M1)

eg ${}_n{C_a} \times {p^a} \times {q^{n - a}}$ , ${\text{B}}\left( {n{\text{, }}p} \right)$

evidence of summing probabilities from $7$ to $12$ (M1)

eg ${\text{P}}\left( {X = 7} \right) + {\text{P}}\left( {X = 8} \right) + \ldots + {\text{P}}\left( {X = 12} \right)$ , $1 - {\text{P}}\left( {X \leqslant 6} \right)$

$0.991248$

$0.991$ A1 N2

[3 marks]

c.i.

finding ${\text{P}}\left( {X = 10} \right)$ (seen anywhere) A1

eg $\left( {\begin{array}{*{20}{c}} {12} \\ {10} \end{array}} \right) \times {0.83^{10}} \times {0.17^2}\,\,\left( { = 0.295952} \right)$

recognizing conditional probability (M1)

eg ${\text{P}}\left( {A\left| B \right.} \right)$ , ${\text{P}}\left( {X = 10\left| {X \geqslant 7} \right.} \right)$ , $\frac{{{\text{P}}\left( {X = 10 \cap X \geqslant 7} \right)}}{{{\text{P}}\left( {X \geqslant 7} \right)}}$

correct working (A1)

eg $\frac{{0.295952}}{{0.991248}}$

$0.298565$

$0.299$ A1 N1

Note: Exception to the FT rule: if the candidate uses an incorrect value for the probability that a flight is on time in (i) and working shown, award full FT in (ii) as appropriate.

[4 marks]

c.ii.

correct equation (A1)

eg $\left( {\begin{array}{*{20}{c}} {20} \\ {19} \end{array}} \right){p^{19}}\left( {1 - p} \right) + {p^{20}} = 0.788$

valid attempt to solve (M1)

eg graph

$0.956961$

$0.957$ A1 N1

[3 marks]

Examiners’ report

[N/A]

c.i.

[N/A]

c.ii.

[N/A]