HOMEWORK (Probability)

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)?

calculate the following:

- P(toothache)
- P(Cavity)
- P(Toothache | cavity)
- P(Cavity | toothache \/ catch)