Mathematical Methods — Question 15

QCAA 2025, Paper 2 · 3 marks

Q15 · 2025 · Technology-activeComplex familiar3 marks

QUESTION 15 (3 marks)

A tour operator offers day cruises off the Queensland coast. They advertise that on any given day, customers have a 65% probability of seeing at least one whale. The tour operator conducts cruises for 40 consecutive days. Let

X

be the binomial random variable representing the outcome — success or failure — over the 40 days. A success is customers seeing at least one whale. Calculate the probability that the number of days with a success will be within one standard deviation of the mean for

X

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WORKED SOLUTION

QCAA guide · typeset solution

3 marks

ANSWER

0.7547

(approximately 75%).

Worked solution

Each day is success or failure, so

X

is binomial with

p=0.65,\qquad n=40.

The mean is

\mu=np=40(0.65)=26.

The standard deviation is

\sigma=\sqrt{40(0.65)(0.35)}=3.0166.

Using integer bounds within one standard deviation of the mean gives 23 to 29 successes. A GDC binomial calculation gives

P(23\le X\le29)=0.7547\approx75\%.

correctly determines the mean

[1 mark]

correctly determines the standard deviation

[1 mark]

determines the required probability

[1 mark]

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Mathematical Methods — Question 15

QUESTION 15 (3 marks)

QCAA guide · typeset solution

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