Mathematical Methods — Question 13

QCAA 2023, Paper 1 · 5 marks

Q13 · 2023 · Technology-freeSimple familiar5 marks

QUESTION 13 (5 marks)

At a certain airport, the departure of one in five international flights is delayed every day. The status of any flight is independent of other flights.

One international flight is selected at random each day for three days. Each selection is recorded as either ‘delayed’ or ‘not delayed’.

State two conditions that make this context suitable for modelling using a binomial random variable.

[2 marks]

Calculate the probability that at least two of the selected flights were delayed.

[3 marks]

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WORKED SOLUTION

QCAA guide · typeset solution

5 marks

ANSWER

a) Two outcomes per trial and constant outcome probabilities; b)

\dfrac{13}{125}

Worked solution

a) One condition is that there are only two outcomes for each selection: ‘delayed’ (success) or ‘not delayed’ (failure), i.e. Bernoulli trials. Another condition is that the probabilities of each outcome do not change in each trial, i.e.

p=\frac15

and

q=\frac45

n=3

p=\frac15

1-p=\frac45

P(\text{at least 2})=P(X=2\text{ or }X=3)

={3\choose2}p^2(1-p)+{3\choose3}p^3={3\choose2}\left(\frac15\right)^2\left(\frac45\right)+\left(\frac15\right)^3=\frac{13}{125}.

correctly states one condition for binomial probability

[1 mark]

correctly states a second condition for binomial probability

[1 mark]

correctly determines the number of trials and the probability of a flight being delayed

[1 mark]

determines a suitable method

[1 mark]

calculates the probability

[1 mark]

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Mathematical Methods — Question 13

QUESTION 13 (5 marks)

QCAA guide · typeset solution

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