Mathematical Methods — Question 12

QCAA 2024, Paper 1 · 6 marks

Q12 · 2024 · Technology-freeSimple familiar6 marks

QUESTION 12 (6 marks)

Each day over a three-day long weekend, a family spins a pointer on a circular board to decide whether they will spend the day at the beach or bushwalking. The circular board consists of three equal sections.

Determine the probability that the family will spend all three days bushwalking.

[1 mark]

b.i)

Determine the following binomial probability, expressed as a fully simplified fraction: exactly two days will be spent at the beach.

[2 marks]

b.ii)

Determine the following binomial probability, expressed as a fully simplified fraction: fewer than three days will be spent at the beach.

[3 marks]

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WORKED SOLUTION

QCAA guide · typeset solution

6 marks

ANSWER

\frac1{27}

; b.i)

\frac49

; b.ii)

\frac{19}{27}

Worked solution

P(\text{3 bushwalking days})=\frac13\times\frac13\times\frac13=\frac1{27}.

b.i) Let

X

be the number of beach days. Then

P(X=2)={3\choose2}\left(\frac23\right)^2\left(\frac13\right)=\frac49.

b.ii)

P(X<3)=1-P(X=3)=1-{3\choose3}\left(\frac23\right)^3\left(\frac13\right)^0=1-\frac8{27}=\frac{19}{27}.

correctly determines the probability

[1 mark]

correctly determines the required binomial probability expression

[1 mark]

determines the probability as a fully simplified fraction

[1 mark]

correctly determines the required method

[1 mark]

determines the required binomial expression for the calculation

[1 mark]

determines the probability as a fully simplified fraction

[1 mark]

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Mathematical Methods — Question 12

QUESTION 12 (6 marks)

QCAA guide · typeset solution

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