A newly opened driving school is running a promotion to advertise their lessons to new drivers with learner permits. For each of the next 150 new drivers who receive their learner permit, a representative of the driving school invites them to flip a coin :

"heads" means the new driver can take driving lessons for free,

"tails" means the new driver will not receive any driving lessons.

LetNbe a random variable denoting the number of new drivers who take the driving lessons.

1. (3 marks) What is the probability distribution ofN?
2. (2 marks) What is the expected value ofN?

The probability that a new driver will progress to passing their driving test without driving lessons is 0.1, while those who have taken the lessons would pass their driving test with probability 0.5.

LetPbe a random variable denoting the number out of all of the new drivers with learner permits who pass their driving test.

1. (6 marks) Derive an expression forE(P|N).
2. (Hint: it may help to writeP=PH+ PT, wherePHis the number who passed the driving test out of those who flipped "heads" (and hence received the lessons), andPTis the number who passed the driving test out of those who flipped "tails" (and missed out on the lessons).
3. (3 marks) Hence calculateE(P).
4. (4 marks) After the promotion is completed you meet a new driver who has passed their driving test. What is the probability that they took the driving lessons?