Scenario. suppose there is a criminal trial. By the presumption of innocence, we
ID: 3172528 • Letter: S
Question
Scenario. suppose there is a criminal trial. By the presumption of innocence, we conclude that the is guilty if all 12 members in a jury agree. Let X be the number of jury members who decides guilty. Let T be the proportion of citizens who would decide guilty if they served for a jury. You may use the below binomial model to answer the following questions: r 0,1,2, where n denotes the total number of jury members (i.e. sample size). 1. If 50% the population would agree that the defendant is guilty, what is the probability for concluding guilty? 2. If 95% of the population would agree that the defendant is guilty, what is the probability for concluding guilty? 3. If 95% of the population would agree that the defendant is guilty, what is the probability for concluding not guilty 4. If 95% of the population would agree that the defendant is guilty, and if we increase the jury size to 20 members, what is the probability for concluding guilty? 5. True or false. The above calculations are valid if one jury member influences other members' decision.Explanation / Answer
1. There are 12 members, that is n=12. The probability of success, p=0.5. The defendent to be proved guilty, all the 12 jury members has to be agreed. Therefore, specific number of success in n=12 trials is x=12. Use binomial distribution to compute the probability.
P(X=12)=12C12(0.5)^12(1-0.5)^0=0.0002
2. Keeping n=12 fixed, and chnaging p to 0.95, the required probability is as follows:
P(X=12)=12C12(0.95)^12(1-0.95)^0=0.5404
3. Probability of concluding non-guilty=1-Probability of concluding guilty
=1-0.5404
=0.4596
4. P(X=20)=20C20(0.95)^20(1-0.95)^0=0.3585 (assume that to prove the defendent guilty, all the jury members has to be agreed, that is n=20 and p=0.95)
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