QCE Mathematical Methods: Exam practice Q17 (2025, Paper 1)

Mathematical Methods — Question 17

QCAA 2025, Paper 1 · 6 marks

Q17 · 2025 · Technology-freeComplex familiar6 marks

QUESTION 17 (6 marks)

In a computer fishing game, a player repeatedly casts their hook into either a blue pond or a red pond. Each fish they catch scores points. In the blue pond, the probability of catching a fish on a cast is

\frac23

and each fish caught scores 10 points. In the red pond, the probability of catching a fish on a cast is

\frac13

and each fish caught scores 15 points. A player has three casts left and needs to score at least 30 points to win. All remaining casts must be in the same pond. It is claimed that the probability of winning if casting in the blue pond is

\frac1{27}

more than the probability of winning if casting in the red pond. Evaluate the reasonableness of the claim.

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

6 marks

ANSWER

The claim is correct:

P(\text{win blue})=\frac8{27}

and

P(\text{win red})=\frac7{27}

, a difference of

\frac1{27}

Worked solution

Let

X

be the number of casts that catch a fish.

Blue pond:

X\sim B(3,\frac23)

. To score at least 30 points requires three successes:

P(X=3)=\left(\frac23\right)^3=\frac8{27}.

Red pond:

X\sim B(3,\frac13)

. To score at least 30 points requires at least two successes:

P(X\ge2)=P(X=2)+P(X=3)

=3\left(\frac13\right)^2\left(\frac23\right)+\left(\frac13\right)^3=\frac6{27}+\frac1{27}=\frac7{27}.

Thus

\frac8{27}-\frac7{27}=\frac1{27}.

The claim is correct.

correctly identifies a method to determine at least 30 points in three casts for the blue pond

[1 mark]

determines the probability of scoring at least 30 points in the blue pond

[1 mark]

correctly identifies a method to determine at least 30 points in three casts for the red pond

[1 mark]

substitutes appropriate values into the chosen method for the red pond

[1 mark]

determines the probability of scoring at least 30 points in the red pond

[1 mark]

evaluates the claim

[1 mark]

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Mathematical Methods — Question 17

QUESTION 17 (6 marks)

QCAA guide · typeset solution

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