**Exercise 1.11**:

Let be the event that after the -th relay we receive the correct bit.

we have a recurrence relation:

Let . We have

Thus becomes

yields

So

**Exercise 1.12**: ( the Monty Hall problem)

Assume without loss of generality that the contestant chose door and Monty opened door to show a goat.

Let be the event that the car is behind door and be the event that Monty opened door .

First we calculate the probabilities before Monty opened door .

.

What is the value of ? If the car is in door (the door the contestant chose), Monty will choose from two remaining door which door to open uniformly at random. So we have .

What is the value of ? If the car is not in door , it can be in either door or with equally probabilities. So .

What is the value of ? if the car is in door , Monty has no other choices than open door (because he cant open door ). So .

Thus

Now we will see how things changed after Monty opened door .

Apply Bayes’ Law yields:

So the contestant will *double* the chance if he or she chooses to* switch doors*.

**Exercise 1.13**:

Let E be the event that the person has the disorder, and B be the event that the result comes back positive.

We have:

Apply Bayes’ Law yields:

So if a person chosen uniformly from the population is tested and the result comes back positive, the probability that the person has the disorder is .

Posted by Anonymous on May 26, 2012 at 2:36 am

Ex1.12: P(B|E3)=1