Let’s Talk Probability
(a) When is P(A∪B)=P(A)+P(B) true? What is the general rule that always holds?
(b) When is P(A∩B)=P(A)∗P(B) true? What is the general rule that always holds?
(c) If A and B are disjoint, are they independent?
2 Aces
Consider a standard 52-card deck of cards:
(a) Find the probability of getting an ace or a red card, when drawing a single card.
(b) Find the probability of getting an ace or a spade, but not both, when drawing a single card.
(c) Find the probability of getting the ace of diamonds when drawing a 5 card hand.
(d) Find the probability of getting exactly 2 aces when drawing a 5 card hand.
(e) Find the probability of getting at least 1 ace when drawing a 5 card hand.
(f) Find the probability of getting at least 1 ace or at least 1 heart when drawing a 5 card hand.
3 Balls and Bins
Throw n balls into n bins.
(a) What is the probability that the first bin is empty?
(b) What is the probability that the first k bins are empty?
(c) Use the union bound to give an upper bound on the probability that at least k bins are empty.
(d) What is the probability that the second bin is empty given that the first one is empty?
(e) Are the events that "the first bin is empty" and "the first two bins are empty" independent?
(f) Are the events that "the first bin is empty" and "the second bin is empty" independent?
