Unizor - Probability - Conditional - Problems 3

We recommend to attempt solving these problems prior to listening to the lecture or reading answers and proofs provided.
Also assume that all probabilities mentioned in the problems are not equal to zero, that is we are excluding impossible events.
Wherever possible, try to represent the problem graphically with a set of elementary events.
The usage of a word "random" assumes equal chances to occurrences of elementary events unless otherwise specified.

Problem 3.1.
We roll two dice. What is the probability that the sum of two rolled numbers equals to 7 if it's known that the first dice shows a number greater than 3?
Answer: 1/6

Problem 3.2.
We roll two dice. What is the probability that the number on the second dice is greater than 3 if it's known that the numbers on both dice are different?
Answer: 1/2

Problem 3.3.
A student is preparing to an exam. Exam includes 100 questions. A student had time to prepare only for the first 70 questions. During the exam students are randomly divided into 2 groups and original 100 questions were also divided among these groups: group 1 got questions from #1 to #50, group 2 got questions from #51 to #100.
Assume that our student can answer correctly any question he had prepared for (those from #1 to #70), but the one he had not prepared for he answers incorrectly.
He randomly picks a question from those given to his group.
(a) What is the probability of answering it correctly if it's known that he was selected into group 1?
(b) What is this probability on condition that he is selected into group 2?
(c) What is the total (unconditional) probability of answering the question correctly?
Answer: 0.7