QCE Vault / Mathematical Methods Discrete random variables — Question 286 Original QCE Vault practice · 5 marks
Browse all questions Original practice exam Report an issue Q286 · Practice question Technology-active Simple familiar 5 marks
QUESTION 286 (5 marks) A ride operator finds that a randomly selected visitor chooses the spinning ride with probability 0.7 0.7 0.7 . Choices by the next 12 12 12 visitors are assumed independent. Let X X X be the number who choose it. a) State the distribution of X X X , including its parameters. [1 mark] b) Determine the probability that at least 10 10 10 choose the ride, to four decimal places. [2 marks] c) Determine the mean and standard deviation of X X X , to two decimal places where necessary. [2 marks] WORKED SOLUTION
Practice marking scheme 5 marks ANSWER a) X ∼ Bin ( 12 , 0.7 ) X\sim\operatorname{Bin}(12,0.7) X ∼ Bin ( 12 , 0.7 ) . b) 0.2528 0.2528 0.2528 . c) Mean 8.4 8.4 8.4 ; standard deviation 1.59 1.59 1.59 . Worked solution
a) There are 12 independent trials with constant success probability 0.7.
b) P ( X ≥ 10 ) = ∑ j = 10 12 ( 12 j ) ( 0.7 ) j ( 0.3 ) 12 − j = 0.25281535 … P(X\geq10)=\sum_{j=10}^{12}\binom{12}{j}(0.7)^j(0.3)^{12-j}=0.25281535\ldots P ( X ≥ 10 ) = ∑ j = 10 12 ( j 12 ) ( 0.7 ) j ( 0.3 ) 12 − j = 0.25281535 … . c) E ( X ) = 12 ( 0.7 ) = 8.4 E(X)=12(0.7)=8.4 E ( X ) = 12 ( 0.7 ) = 8.4 and σ = 12 ( 0.7 ) ( 0.3 ) = 2.52 \sigma=\sqrt{12(0.7)(0.3)}=\sqrt{2.52} σ = 12 ( 0.7 ) ( 0.3 ) = 2.52 . Exact answers unless rounding is specified; equivalent forms accepted.
a) Gives the binomial distribution and parameters.
[1 mark] b) Uses the correct inclusive upper tail.
[1 mark] b) Gives the rounded probability.
[1 mark] c) Gives the standard deviation.
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