The internal evaluation for Economics course in an Engineering programme is based on the score of four quizzes. Rahul has secured 70, 90 and 80 in the first three quizzes. The fourth quiz has ten True-False type questions, each carrying 10 marks. What is the probability that Rahul’s average internal marks for the Economics course is more than 80, given that he decides to guess randomly on the final quiz?

Rahul has to score either 90 or 100 marks in the fourth quiz in order to have average more than 80. 

So,there will be two cases:

Case 1: If Rahul scores 90 marks

Then 9 out of 10 will be correct and those 9 correct answers can be in any order. So, total ways of arranging them is $$\frac{10!}{9!}$$

And the probability of choosing either a right answer or wrong answer is $$\frac{1}{2}$$

Hence, the probability of getting 9 answers correct is: $$\frac{10!}{9!}$$  x $$(\frac{1}{2})^{10}$$

Case 2: If Rahul scores 100 marks

Then 10 out of 10 will be correct. So, total ways of arranging them is $$\frac{10!}{10!}$$ = 1

And the probability of choosing either a right answer or wrong answer is $$\frac{1}{2}$$

Hence, the probability of getting all 10 answers correct is: $$(\frac{1}{2})^{10}$$

So, the final probability is $$\frac{10!}{9!}$$  x $$\frac{1}{2^{10}}$$ + $$\frac{1}{2^{10}}$$ = $$\frac{11}{1024}$$