QCE Vault / Specialist Maths Exponential random variables — Question 212 Original QCE Vault practice · 5 marks
Browse all questions Original practice exam Report an issue Q212 · Practice question Technology-active Complex familiar 5 marks
QUESTION 212 (5 marks) Two independent sensors have exponential lifetimes, each with mean 5 hours.
a) Find the probability both survive at least 5 hours.
[2 marks] b) Find the probability at least one survives at least 5 hours. Give four decimal places.
[3 marks] WORKED SOLUTION
Practice marking scheme 5 marks ANSWER (a) e − 2 ≈ 0.1353 e^{-2}\approx0.1353 e − 2 ≈ 0.1353 . (b) 1 − ( 1 − e − 1 ) 2 ≈ 0.6004 1-(1-e^{-1})^2\approx0.6004 1 − ( 1 − e − 1 ) 2 ≈ 0.6004 . Worked solution
(a) Each rate is 1/5, so individual survival to 5 is exp(-1). Independence gives joint survival exp(-2).
(b) Use the complement that both fail. Each failure probability is 1-exp(-1), so at least one survivor has probability 1 − ( 1 − e − 1 ) 2 1-(1-e^{-1})^2 1 − ( 1 − e − 1 ) 2 . This avoids double-counting two survivors. Equivalent mathematically justified methods accepted.
Exact answers unless a decimal accuracy is specified.
Part a: Find individual survival.
[1 mark] Part a: Multiply independently and report 0.1353.
[1 mark] Part b: Identify the complement.
[1 mark] Part b: Multiply independent failure probabilities.
[1 mark] Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.
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