1. (General probability)
Consider two events A and B such that 0 < P(B) < 1. For each of the following statements, decide whether they are always true for any such A and B, always false or could be either.
a. P(A ∪ B) ≥ P(A) + P(B)
b. P(A|B) ⋅ P(A) = P(A ∩ B)
c. P(A ∪ B|B) ≥ P(A|B)
d. P(A ∩ B|B) + P(A ∩ Bc |B) = 1
e. P(A|B) + P(Ac |B) < 1
f. A⊥B ⇒ P(A|B) = P(A|Bc)
g. A⊥B ⇒ P(A ∪ B) = P(A) + P(B)
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2. (Events and probability)
a. Let E1, E2 be two events. Prove that P(E1\ E2) = P(E1) - P(E2) implies P(E2\ E1) = 0.
b. A red, a blue and a green die are rolled simultaneously.
i. What is the probability that the sum of the 3 values is a prime number?
ii. What is the probability of a doublet (2 of the 3 values are equal but the third value is different)?
iii. What is the probability that all 3 values are different and the value of the red die is higher than the other two?
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3. (Counting and probability) Let Σ = {a,c,e,n,s}. Suppose we choose a 4-letter word at random from Σ4. What is the probability that the letters of the word are in alphabetical order (e.g. as in aces or cees but not in sees)? [show answer]
4. (Probability of sequences)
Alice and Bob repeatedly toss a coin (outcome H – head, or T – tails) until either Alice's or Bob's winning sequence is observed. What is the probability for Alice to win if
a. Alice's winning sequence is HTH and Bob's is HHH?
b. Alice's winning sequence is HTH and Bob's is THT?
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5. (Conditional Probability)
a. Prove that
P(A1 ∩ A2 ∩ … ∩ An) = P(A1) ⋅ P(A2|A1) ⋅ P(A3|A1 ∩ A2) ⋅ … ⋅ P(An|A1 ∩ A2 ∩ … ∩ An−1)
b. Consider three urns: Urn 1 contains one red and two black marbles, Urn 2 contains three red and four black marbles, Urn 3 contains two red and two black marbles. One urn is selected at random and then two
marbles are randomly drawn from that urn without replacement. Given that these two marbles are red, what is the probability that Urn 2 was chosen?
[show answer]
6. Challenge Exercise
a. Jamie and Charlie have two kids, one of which is a girl. Assume that the probability of each gender is ½. What is the probability that the other kid is also a girl?
There are three cards in a hat. One card is red on both sides, one card is blue on both sides, and one card is red on one side and blue on the other. You draw one card at random, put it down on the table and then observe that the side of the card facing up is red. What is the probability that the other side is red too?
