Homework 1 Solution - Spring 2008 UGBA 143 Homework 1...

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Spring 2008, UGBA 143 Homework 1 Solution 1. (Pooled Testing) General Set-up: - Individual Testing: N tests (N = number of donors). - Pooled Testing: at least 1 test, but possibly N more tests. - p = probability of a given donor to be HIV-positive - Note that p here is also the percentage of HIV-positive donors in the population.  - Probability that a pool of N tests contains NO positive tests: (1-p) - Probability that the population contains AT LEAST one positive test: 1-(1-p) N - Expected number of tests for Pool Testing: 1 + N(1-(1-p) N - Therefore, pooled testing is unfavorable if   1 + N(1-(1-p) N ) < N  a) N = 5, P = 0.25 1+5*(1-0.75 5 ) = 4.81 < 5   Pool Test is better b) N = 5, P = 0.4 1+5*(1-0.6 5 ) = 5.61 > 5  
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Unformatted text preview: Individual Test is better c) N = 10, P = 0.25 1+10*(1-0.75 10 ) = 10.43 > 10 Individual Test is better d) Refer to set-up, any discussion similar is fine. 2. (Martingale System) a) Expected Monetary Gain (Profit) from making the bold bet = 1000*0.5 + (-1000)*0.5 = $0 EV= $0 0.5 0.5 +1000-1000 b) Cautious bet of $10 iii) Cautious bet of $100 b) Notice that any bet of the form has an expected profit of $0, which is why the above two trees all traces back to $0. Notice also that, for more cautious bets, the probability of winning is higher, but the amount of profit is lower (lower risk/value strategy for more risk-averse players) EV= $0 0.5 0.5 +X-X...
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  • Spring '08
  • XuanmingSu
  • Probability theory, Cautious bet

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