- a)
- [1 P] Prove the conditionalized version of Bayes' rule:
- b)
- [1 P] Consider the domain of dealing 5-card poker hands from a standard deck of 52 cards, under the assumption that the dealer is fair.
- How many atomic events are there in the joint probability distribution (i.e., how many 5-card hands are there)?
- What is the probability of each atomic event?
- What is the probability of being dealt a royal straight flush? Four of a kind?

- c)
- [1 P] Suppose you are given a bag containing
unbiased coins. You are told that
of these coins are normal, with heads on one side and tails on the other, whereas one coin is a fake, with heads on both sides:
- Suppose you reach into the bag, pick out a coin uniformly at random, flip it, and get a head. What is the (conditional) probability that the coin you chose is the fake coin?
- Suppose you continue flipping the coin for a total of
times after picking it and see
heads. Now what is the conditional probability that you picked the fake coin?
- Suppose you wanted to decide whether the chosen coin was fake by flipping it times. The decision procedure returns if all flips come up heads, otherwise it returns . What is the (unconditional) probability that this procedure makes an error?

- Suppose you reach into the bag, pick out a coin uniformly at random, flip it, and get a head. What is the (conditional) probability that the coin you chose is the fake coin?
- d)
- [1 P] After your early checkup, the doctor has bad news and good news. The bad news is that you tested positive for a serious disease and that the test is 99% accurate (i.e., the probability of testing positive when you do have the disease is 0.99, as is the probability of testing negative when you don't have the disease). The good news is that this is a rare disease, striking only 1 in 10000 people of your age. Why is it good news that the disease is rare? What are the chances that you actually have the disease?
- e)
- [1* P]
The police investigates a crime scene and want to find out which car the criminals used. They find traces of a special car varnish which is often found on Italian sports cars. 80 % of all Ferraris, but only 1 % of all cars in the country have this varnish. However, only 0.01 % of all cars in the country are Ferraris. On the other hand, 5 % of all cars in the country are VW Golf, but this varnish is used for only 2 % of all Golfs. Use Bayes' rule to calculate the likelihood for both cars and decide which car the police should look for. Exactly define all symbols that you use.

The police later also finds tire tracks. This kind of tires is used in 75 % of all Ferraris, but only in 1 % of all VW Golf. Update the probabilities for both car types. Which car should the police look for now, and how much more likely is this car type than the other one? Write down all your assumptions and exactly define all symbols that you use.