1/Each round Alice, Bob, and Carlos play a game, Alice wins with probability 0.5

Question

1/Each round Alice, Bob, and Carlos play a game, Alice wins with probability 0.5, Bob wins with probability 0.3, and Carlos wins with probability 0.2, independent of other rounds. If they play 8 rounds, (a) what is the probability that Alice wins 3 rounds, Bob wins 4 rounds , and Carlos wins 1 round? (b) what is the probability that Carlos wins at least once? 2/ (False Positive) A company believes 95% of its jobs applicants are trustworthy. The company gives everyone a polygraph test, asking “have you ever stolen anything from your workplace”, to which all applicants naturally answer no. The company then uses a polygraph (lie de- tector) to see if they are trustworthy. Suppose when a person lies, the polygraph detects the lie with probability 65%, but incorrectly identifies 15% of true statements as lies. What is the probability that a job applicant rejected under suspicion of dishonesty was actually trustworthy?

Explanation / Answer

a)probability that Alice wins 3 rounds, Bob wins 4 rounds , and Carlos wins 1 round

=(8!/(3!*4!*1!))*(0.5)3*(0.3)4*(0.2) =0.0567

b) probability that Carlos wins at least wins none) =1-(1-0.2)8 =0.8322

2)probability that a job applicant rejected under suspicion of dishonesty =P(trustworthy and rejected+not trustwoerthy and rejected) =0.95*0.15+0.05*0.65=0.175

therefore  probability that a job applicant rejected under suspicion of dishonesty was actually trustworthy

=P(trustworthy and rejected)/P(rejected) =0.95*0.15/0.175=0.814286

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