Stat 13C - Answers to the Sample Final - F08

# Stat 13C - Answers to the Sample Final - F08 - Stat 13C F J...

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Stat 13C Answers to the Sample Final Fall 2008 F. J. Samaniego 1. Drawing a probability tree is helpful, though not mandatory. Let W 1 be the event that Penn State wins the Fiesta Bow and let W 2 be the event that Iowa wins the Rose Bowl. The tree you might have drawn has four paths. The two you want (using Ls for losses) are W 1 L 2 and L 1 W 2 . The probability of winning exactly one bet is (.4)(.3) + (.6)(.7) = .54. 2. The “box” method will help here. P(M c |C c ) = P(M c C c ) / P(C c ) = (.4) / (.6) = 2/3. 3. Definitely a tree problem. Let G 1 be the unloaded gun and G 2 the other. Let B be the event that a bullet comes out when the gun is fired. P(B) = (1/2)(0) + (1/2)(1/3) = 1/6. 4. A Bayes problem. P(G 1 |B c ) = P(G 1 B c ) / P(B c ) = 3/5. 5. X ~ B(6, 1/3). P(X = 4) = 4 6 (1/3) 4 (2/3) 2 = .0823. 6. Approximate p as .3. Then, approximately, X ~ B(10, .3). From Table 1, P(X ≥ 5) = 1 - .850 = .150. 7. Let X ~ B(72, 1/3); np = 24 and nq = 16, so normal approximation is valid. Let Y ~ N(24, 16). Then P(X ≥ 30) ≈ P(Y ≥ 29.5) = P(Z ≥ 1.38) = .0838.

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