Question

5.) A jar contains 50 type 1 coins and 50 type 2 coins. Type 1 coins have probability 2/3 of landing on heads, while type 2 coins have probability 1/3 of landing on heads. If a coin is chosen at random from the jar and tossed 3 times, what is the probability that it lands on heads at least 2 times? 6) Suppose that an HIV-1 test has a false positive rate of 0.1% and a false negative rate of 0.5%. If an individual with no known risk factors from a population with an HIV-1 prevalence of 0.0001 is tested twice and receives a positive result both times, what is the probability that they are actually infected? State any additional assumptions that you use to derive your answer.

Answer 1

5)P(land on head at least twice) =P(from Jar 1 and land on heads at least twice)+P(from Jar 2 and land on heads at least twice)

=(1/2)*(³C₂(2/3)²(1/3)+³C₃(2/3)³(1/3)⁰)+(1/2)*(³C₂(1/3)²(2/3)+³C₃(1/3)³(2/3)⁰)=0.5

6)

we assume that consecutive tests are independent

P(infected given tested positive twice) =P(infected and tested positive twice)/P(tested positive twice)

=(0.0001*0.995*0.995)/(0.0001*0.995*0.995+0.9999*0.001*0.001)=0.9900