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Problem set 3: 4. An urn holds 50 marbles of which five marbles are marked. The marbles are divided up uniformly at randominto five groups. What is
4. An urn holds 50 marbles of which five marbles are marked. The marbles are divided up uniformly at randominto five groups. What is the probability that each group holds exactly one of the marked marbles?
5. An urn holds n (for some n ∈ N) marbles of which two are marked. The marbles are divided uniformly atrandom into k groups for some k|n (i.e., n is a multiple of k, so the groups each hold the same number ofmarbles). What is the probability that the two marked marbles are in the same group?
6. A slot machine takes a coin and either returns two coins (the player wins) with probability p (for somep ∈ (0, 1)) or keeps the coin (the player loses) with probability 1 − p. Unfortunately p is unknown. It isknown that the slot machine manufacturer creates three different types of machines, with win probabilitiesp = 1/4 (favorable to the house), p = 1/2 (fair), and p = 3/4 (favorable to the player), and that the currentmachine in use is equally likely to be any one of these three types. You play the slot machine n times andrecord the number of plays, k ∈ {0, . . . , n} in which you win. Give the probability that the machine is fairas a function of (k, n).
7. Two distinguishable fair n-sided dice are rolled, where n is an even number. Define events: A is the eventthat the first die shows an even number, B is the event that the second die shows an even number, and Cis the event that the two dice show the same number. Are (A, B, C) pairwise independent? Are (A, B, C)independent?
8. You hold two coins: a fair coin and a two-headed coin. You select a coin uniformly at random and then,without looking at the coin you have selected, flip it n times in a row. Define events A1, . . . , An where Aiisthe event that flip i ∈ [n] is a head, and define A = A1 ∩ · · · ∩ An as the event that all n flips shows heads.Define B as the event that the fair coin is tossed. Find P(A) as a function of n. Find P(B|A) as a functionof n.
