Q-1: For each of the following statements, either prove it is true or give a counterexample.
- If P(a | b,c) = P(b | a,c), then P(a | c) = P(b | c).
- If P(a | b,c) = P(a), then P(b | c) = P(b).
- If P(a | b) = P(a), then P(a | b,c) = P(a|c).
Q-2: Consider the set of all possible five-card poker hands dealt fairly from a standard deck of fifty-two cards.
- How many atomic events are there in the joint probability distribution (i.e., how many five-card hands are there)?
- What is the probability of being dealt a royal straight flush? a stright flush? a stright?
- What is the probability of being dealt a full house (three of a kind and two of another kind)?
Q-3: Consider the following full joint distrubition:
calculate the following:
- P(toothache)
- P(Cavity)
- P(Toothache | cavity)
- P(Cavity | toothache \/ catch)
Q-4: Consider two medical tests, A and B, for a virus,
Test A is 95% effective at recognizing the virus when it is present,
but has a 10% false positive rate (indicating that the virus is
present, when it is not). Test B is 90% effective at recognizing the
virus, but has a 5% false positive rate. The two tests use independent
methods of identifying the virus. The virus is carried by 1% of all
people. Say that a person is tested for the virus using only one of the
tests, and that test comes back positive for carrying the virus. Which
test returning positive is more indicative of someone really carrying
the virus? Justify your answer mathematically.