Answer Happy

6) An innocent looking yet sadistic professor named Mr. M.P. Hola wants to ask a series of true and false questions for exam. He has decided that the average probability of a question having the 3 answer as "True" should be p. He now wants a mathematical proof of the fact that p= 0.5 will make these answers hardest to guess. Do that for him. Also, being Mr. Hola, he wants to leak a fake answer key for this exam, where the provided answer will match the actual answer of a question with a probability q. Help Mr. Hola choose q (using a mathematical proof) so that looking at an answer in the leaked answer key does not convey any information about the true answer. Now, based on this discussion, consider a digital communication transmitting and receiving binary symbols, and describe the worst possible probability of error that this system can have. {5+8+3=16 Marks}