6.042/18.] Mathematics for Computer Science April 27, 2005 Srini devadas and Eric Lehman Notes for recitation 19 roblem 1. Suppose that you flip three fair mutually independent coins. Define the fol- lowing events Let A be the event that the first coin is heads Let b be the event that the second coin is heads Let c be the event that the third coin is heads Let d be the event that an even number of coins are heads (a)Are these events pairwise independent? Solution. The sample space consists of eight, equiprobable outcomes [H, T]3=HHH, HHT, HTH, HTT, THH, THT, TTH, TTT) Each sequence of three symbols specifies the outcomes of the three coin tosses The first three events(A, B, and C) are pairwise independent, since they are mu- tually independent. all that remains is to check that each of these is independent D. By symmetry, we need only check just one of the three, say A Pr(A)=号 Pr(D)= Pr(HHT, HTH, THH, TTT)) Pr(AnD)=Pr(HHT, HTH= Therefore, Pr(An D)= Pr (A). Pr(D), and so these events are independent. We conclude that all four events are pairwise independent (b) are these events three-way independent? That is does Pr(X∩Y∩Z)=Pr(X)Pr(Y)Pr(z) always hold when X, Y, and Z are distinct events drawn from the set (A, B, C, D)? Solution. Because the coin tosses are mutually independent, we know Pr(A∩B∩C)=Pr(A)Pr(B)Pr(6.042/18.062J Mathematics for Computer Science April 27, 2005 Srini Devadas and Eric Lehman Notes for Recitation 19 Problem 1. Suppose that you flip three fair, mutually independent coins. Define the fol￾lowing events: • Let A be the event that the first coin is heads. • Let B be the event that the second coin is heads. • Let C be the event that the third coin is heads. • Let D be the event that an even number of coins are heads. (a) Are these events pairwise independent? Solution. The sample space consists of eight, equiprobable outcomes: {H, T}3 = {HHH, HHT, HT H, HT T, T HH, T HT, T T H, T T T} Each sequence of three symbols specifies the outcomes of the three coin tosses. The first three events (A, B, and C) are pairwise independent, since they are mu￾tually independent. All that remains is to check that each of these is independent of D. By symmetry, we need only check just one of the three, say A: Pr (A) = Pr (D) 1 2 1 = Pr ({HHT, HT H, THH, T T T}) = 1 2 Pr (A ∩ D) = Pr ({HHT, HT H}) = 4 Therefore, Pr (A ∩ D) = Pr (A) · Pr (D), and so these events are independent. We conclude that all four events are pairwise independent. (b) Are these events three­way independent? That is, does Pr (X ∩ Y ∩ Z) = Pr (X) · Pr (Y ) · Pr (Z) always hold when X, Y , and Z are distinct events drawn from the set {A, B, C, D}? Solution. Because the coin tosses are mutually independent, we know: Pr (A ∩ B ∩ C) = Pr (A) · Pr (B) · Pr (C)