Answer to Question #111494 in Statistics and Probability for bob

Question

There are A, B, and C types of problem set, and each type of problem set contains 3000, 2000, 5000
problems, respectively. Suppose you can solve A-type problem with 90% probability, B-type problem with
20% probability, and C-type problem with 60% probability. To pass graduation exam, you must solve both
of two questions randomly chosen from the total problem set (out of 10,000 problems). Answer the
following questions:

Answer 1

How many questions can I solve?

$$0.9\times3000+0.2\times2000+0.6\times5000=6100$$

The probability that I pass the graduation exam

$$P(I\ pass)={6100 \over 10000}\times{6099 \over 9999}\approx0.372076$$

The probability that I have solved one or more A-type questions when you passed the exam

$$P(A|I\ pass)={P(A\cap I\ pass) \over P(I \ pass)}$$

$$P(A\cap I\ pass)={2700 \over 10000}\times{3400 \over 9999}+{3400 \over 10000}\times{2700 \over 9999}+$$

$$+{2700 \over 10000}\times{2699 \over 9999}\approx0.261845$$

$$P(A|I\ pass)={0.261845\over 0.372076}\approx0.7215$$