A master student is planning to take a qualifying exam in the coming summer. She has three chances. The first one is in June. If she fails the 1st one, then she can have a 2nd try in July, and if she also fails that one, she will take the 3rd one in August. If she fails all three exams, then she is not qualified. The chance of passing the 1st one is 0.5, the chance of passing the 2nd one is 0.7, and the chance of passing the 3rd one is 0.8. And these chances are independent. a) What is the probability that she passes the qualification? b) Given that she passes the qualification, what is the conditional probability that she passes on the second try?

Answer:Step-by-step explanation:Given[tex]P_1=[/tex]Probability of clearing exam in 1 st attempt=0.5[tex]P_2=[/tex]Probability of clearing exam in 2 nd attempt=0.7[tex]P_3=[/tex]Probability of clearing exam in 3 rd attempt=0.8Probability that she passes the exam[tex]=P(1 st\ attempt)+P(1\ fail)\cdot P(2\ pass)+P(1\ fail)\cdot P(2\ fail)\cdot P(3 rd pass)[/tex][tex]P=0.5+0.5\times 0.7+0.5\times 0.3\times 0.8=0.97[/tex](b)P(Pass qualification on 2nd try|passes qualification)P(Pass qualification on 2nd try|passes qualification)[tex]=\frac{P(fail\ on\ 1 )\cdot P(pass\ in\ 2nd)}{P(passes)}[/tex]P(Pass qualification on 2nd try|passes qualification)[tex]=\frac{0.5\times 0.7}{0.97}=0.3608[/tex]