QCE Vault / Mathematical Methods Discrete random variables — Question 288 Original QCE Vault practice · 4 marks
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QUESTION 288 (4 marks) A puzzle room offers two rounds. In each round, a team attempts three locks. Each lock opens with probability 0.4 0.4 0.4 , independently of all other attempts. A round is completed only if all three locks open. a) Determine the probability of completing one round.
[1 mark] b) Determine the probability of completing at least one of the two rounds.
[2 marks] c) Let Y Y Y be the number of completed rounds. Determine E ( Y ) E(Y) E ( Y ) . [1 mark] WORKED SOLUTION
Practice marking scheme 4 marks ANSWER a) 0.064 0.064 0.064 . b) 0.123904 0.123904 0.123904 . c) 0.128 0.128 0.128 . Worked solution
a) All three successes are required: q = ( 0.4 ) 3 = 0.064 q=(0.4)^3=0.064 q = ( 0.4 ) 3 = 0.064 . b) The probability of completing neither round is ( 1 − q ) 2 (1-q)^2 ( 1 − q ) 2 . The required probability is 1 − ( 0.936 ) 2 = 0.123904 1-(0.936)^2=0.123904 1 − ( 0.936 ) 2 = 0.123904 . c) Y ∼ Bin ( 2 , 0.064 ) Y\sim\operatorname{Bin}(2,0.064) Y ∼ Bin ( 2 , 0.064 ) , so E ( Y ) = 2 ( 0.064 ) = 0.128 E(Y)=2(0.064)=0.128 E ( Y ) = 2 ( 0.064 ) = 0.128 . Exact answers unless rounding is specified; equivalent forms accepted.
a) Uses the product of three independent probabilities.
[1 mark] b) Uses the complement of two failed rounds.
[1 mark] b) Calculates the probability.
[1 mark] Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.
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