Lecture-10a

Lecture-10a - f ω X ω 1 f ω 1 X ω 2 f ω 2 X ω 3 f ω 3 =(0 f ω(2 f ω 1(10 f ω 2(50 f ω 3 =(2(1 100(10(1 1000(50(1 10000 = 035 = 3 1 2

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3.5:35. Extension a. An experiment has probability p of success, q of failure and 1 - p - q of neither. If independent trials are repeated over and over until either success or failure is obtained, what is the probability of an ultimate success? Solution. Let S be the event of success on the Frst trial, let F be the event of getting failure on Frst trial, let N be the event of getting neither on the Frst trial. Let E be the event of getting a success before a failure. Then P ( E ) = P ( E | S ) · P ( S ) + P ( E | F ) · P ( F ) + P ( E | N ) · P ( N ) = 1 · p + 0 · q + P ( E ) · (1 - p - q ) = p + P ( E ) · (1 - p - q ) . Thus, 0 = p - P ( E )( p + q ) , so P ( E ) = p p + q . Extension b. In a lottery with a very large number of tickets, 1 in 100 tickets pay $2, 1 in 1000 tickets pay $10, 1 in 10000 tickets pay $50. What is a ticket worth? Solution. Let X ( ω i ) be the payo± for the outcome of getting a ticket of type i , where i = 0 if the ticket is a looser, i = 1 if it wins $2, i = 2 if it wins $10 and i = 3 if it wins $50. Thus, X ( ω 0 ) = 0, X ( ω 1 ) = 2, etc. Let f be the pmf. E ( X ) = X ( ω 0 )
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Unformatted text preview: f ( ω ) + X ( ω 1 ) f ( ω 1 ) + X ( ω 2 ) f ( ω 2 ) + X ( ω 3 ) f ( ω 3 ) = (0) f ( ω ) + (2) f ( ω 1 ) + (10) f ( ω 2 ) + (50) f ( ω 3 ) = (2)(1 / 100) + (10)(1 / 1000) + (50)(1 / 10000) = . 035 = 3 1 2 cents Part 2. Now suppose that on Mardi Gras, 1 in ten tickets wins two additional tickets in the same lottery. What is a ticket on Mardi Gras worth? Solution. Let ω 4 be the event of getting the bonus Mardi Gas ticket. If E ( X ) = e , then X ( ω 4 ) = 2 e . e = E ( X ) = X ( ω ) f ( ω ) + X ( ω 1 ) f ( ω 1 ) + X ( ω 2 ) f ( ω 2 ) + X ( ω 3 ) f ( ω 3 ) + X ( ω 4 ) f ( ω 4 ) = (0) f ( ω ) + (2) f ( ω 1 ) + (10) f ( ω 2 ) + (50) f ( ω 3 ) + (2 e ) f ( ω 4 ) = (2)(1 / 100) + (10)(1 / 1000) + (50)(1 / 10000) + (2 e )(1 / 10) e = . 035 + e/ 5 e = . 04375 = 4 3 8 cents 1...
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This note was uploaded on 11/29/2011 for the course MATH 3355 taught by Professor Britt during the Spring '08 term at LSU.

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