Test 2 Solutions

Test 2 Solutions - STAT 230 Name: Test 2 May 31, 2006 ID:...

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STAT 230 Test 2 May 31, 2006 3:30 – 4:10 pm Name: ID: UWUserid: Circle your section : Section 01 (1:30pm) Section 02 (11:30am) Section 03 (9:30am) 1. In the final stage of a game show, two contestants take it in turns to answer questions. Each question is picked at random from a large stack, of which 40% relate to sports and the remaining 60% are about history. The first contestant, Angela, has a 70% chance of answering a sports question correctly, and a 50% chance of answering a history question correctly. The second contestant, Brian, has a 20% chance of answering a sports question correctly, and a 90% chance of answering a history question correctly. If a contestant fails to answer a question correctly, the question is discarded and a new question picked to ask the other contestant. The cycle continues until one of the contestants answers correctly. The first question is directed at Angela, and the first contestant to answer a question correctly wins the game. Assume that contestants’ answers to different questions are independent. [2] (a) Show that at each turn, Angela has a probability of 0.58 of answering her question correctly. P (correct) = P (correct | sports) P (sports) + P (correct | history) P (history) = 0 . 4 × 0 . 7 + 0 . 6 × 0 . 5 = 0 . 58 [2] (b) Show that at each turn, Brian has a probability of 0.62 of answering his question correctly P (correct) = P (correct | sports) P (sports) + P (correct | history) P (history) = 0 . 4 × 0 . 2 + 0 . 6 × 0 . 9 = 0 . 62 [2] (c) Let round 1 begin when Angela is asked her first question, and round 2 when she is asked her second question. In general, round n begins when Angela is asked her n th question, and ends before she is asked her ( n + 1) th question. Find the probability that Angela wins on round 3. P (A wins on round 3) = P (both answer wrong on round 1) × P (both wrong on round 2) × P (A right on round 3) = (0 . 42 × 0 . 38) 2 × 0 . 58 = 0 . 01477 [2] (d) Find the probability that Brian wins on round 3
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P (B wins on round 3) = P (both answer wrong on round 1) × P (both wrong on round 2) × P (A wrong on round 3) × P (B right on round 3) = (0 . 42 × 0 . 38) 2 × 0 . 42 × 0 . 62 = 0 . 00663 [2] (e) Find the probability that the game ends on round 3. P
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Test 2 Solutions - STAT 230 Name: Test 2 May 31, 2006 ID:...

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