QCE Vault / Mathematical Methods Discrete random variables — Question 287 Original QCE Vault practice · 5 marks
Browse all questions Original practice exam Report an issue Q287 · Practice question Technology-free Complex familiar 5 marks
QUESTION 287 (5 marks) A spinner pays a prize W W W dollars. The prizes 0 0 0 , 4 4 4 and 10 10 10 occur with probabilities 0.5 0.5 0.5 , 0.3 0.3 0.3 and 0.2 0.2 0.2 , respectively. A player pays a fee d d d dollars and receives net gain G = W − d G=W-d G = W − d . a) Determine the fee that makes E ( G ) = 0 E(G)=0 E ( G ) = 0 . [2 marks] b) Determine Var ( G ) \operatorname{Var}(G) Var ( G ) for this fee. [2 marks] c) Determine the probability of a positive net gain.
[1 mark] WORKED SOLUTION
Practice marking scheme 5 marks ANSWER a) $d=$3.20 . b ) . b) . b ) 14.56 d o l l a r s s q u a r e d . c ) dollars squared. c) d o l l a r ss q u a r e d . c ) 0.5$. Worked solution
a) E ( W ) = 4 ( 0.3 ) + 10 ( 0.2 ) = 3.2 E(W)=4(0.3)+10(0.2)=3.2 E ( W ) = 4 ( 0.3 ) + 10 ( 0.2 ) = 3.2 . Since E ( G ) = E ( W ) − d E(G)=E(W)-d E ( G ) = E ( W ) − d , choose d = 3.2 d=3.2 d = 3.2 . b) E ( W 2 ) = 16 ( 0.3 ) + 100 ( 0.2 ) = 24.8 E(W^2)=16(0.3)+100(0.2)=24.8 E ( W 2 ) = 16 ( 0.3 ) + 100 ( 0.2 ) = 24.8 , so Var ( G ) = Var ( W ) = 24.8 − 3.2 2 = 14.56 \operatorname{Var}(G)=\operatorname{Var}(W)=24.8-3.2^2=14.56 Var ( G ) = Var ( W ) = 24.8 − 3. 2 2 = 14.56 . c) Positive gain occurs for W = 4 W=4 W = 4 or 10 10 10 , with probability 0.3 + 0.2 = 0.5 0.3+0.2=0.5 0.3 + 0.2 = 0.5 . Exact answers unless rounding is specified; equivalent forms accepted.
a) Calculates the expected prize.
[1 mark] a) Sets the fee equal to that expectation.
[1 mark] b) Calculates E(W squared).
[1 mark] b) Gives the variance, unchanged by the fixed fee.
[1 mark] c) Finds the probability of positive gain.
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