2025 VCE Maths Methods Mini Test 6
Number of marks: 14
Reading time: 3 minutes
Writing time: 21 minutes
Section B – Calculator Allowed
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In questions where a numerical answer is required, an exact value must be given unless otherwise specified.
• In questions where more than one mark is available, appropriate working must be shown.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
(14 marks)
The time taken for a driver to travel to work each day, in minutes, is modelled by a continuous random variable \(T\) with probability density function
\[ f(t) = \begin{cases} \dfrac{12}{1215000}(t - 29)(59 - t)^3 & 29 \leq t \leq 59 \\ 0 & \text{otherwise} \end{cases} \]
a.
i. Find the mean time taken, in minutes, for the driver to travel to work each day. 1 mark
ii. Find the standard deviation of the time taken, in minutes, for the driver to travel to work each day. 2 marks
b. The driver allows \(k\) minutes to travel to work each day. If the journey takes longer than \(k\) minutes, the driver will be late. Whether the driver is late on a particular day is independent of whether they are late on any other day.
i. If \(k = 47\), write a definite integral to show that the probability of the driver being late is 0.08704. 1 mark
ii. If \(k = 47\), find the probability that the driver will be late on at least one day in a five-day working week. Give your answer correct to four decimal places. 2 marks
iii. For \(k = 47\), let \( \hat{P} \) be the proportion of days the driver is late in any five-day working week. Find \(\Pr(0.4 \leq \hat{P} \leq 0.6)\) correct to four decimal places. 2 marks
iv. Find the integer \(k\) such that the probability, correct to one decimal place, of the driver being late at least once in any five-day working week is 0.2. 2 marks
c. At a given traffic light, the wait time is modelled by a normal distribution with a mean of 2.5 minutes and a standard deviation of \(\sigma\) minutes.
i. If \(\sigma = 0.6\), find the probability that the wait time will be less than 3.5 minutes. Give your answer correct to two decimal places. 1 mark
ii. Find the value of \(\sigma\) such that there is a 2% chance of a wait time longer than 3.5 minutes. Give your answer correct to two decimal places. 1 mark
d. The driver passes through three traffic lights (\(A\), \(B\) and \(C\)) on their journey to work.
The probability of each traffic light being red is shown in the table below.
Let \(Y\) be the random variable representing the number of traffic lights that are red on the driver's journey to work. Assume that each traffic light being red is independent of any other traffic light being red.
Complete the following table for the probability distribution of \(Y\). 2 marks
End of examination questions
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