We toss three fair coins simultaneously and independently. If the outcomes of all coin tosses are the same, we win; otherwise, we lose. Let A be the event that the first coin and second coin come up heads, B be the event that the second coin and third coin come up heads, and C be the event that we win. Determine whether the below events are true or false. A. Events A and B are not independent. B. Events A and C are independent. C. Events A and B are conditionally independent given C. D. Events A and C are conditionally independent given B. E. The probability of winning is 3

Answer:A. TRUE B. FALSE C. FALSE D. FALSE E. FALSEStep-by-step explanation:The sample space is S = {HHH, HHT, HTH, THH, TTT, TTH, THT, HTT}, and we have the following events. A = {HHH, HHT} B = {THH, HHH} C = {HHH, TTT} A. P(A\cap B) = P({HHH}) = 1/8, P(A) = 1/4, P(B) = 1/4, P(A)P(B) = (1/4)(1/4) = 1/16. Because [tex]P(A\cap B) = 1/8 \neq 1/16 = P(A)P(B)[/tex], we have that A and B are not independent. B. P(A\cap C) = P({HHH}) = 1/8, P(A) = 1/4, P(C) = 1/4, P(A)P(C) = (1/4)(1/4) = 1/16. Because [tex]P(A\cap C) = 1/8 \neq 1/16 = P(A)P(C)[/tex], we have that A and C are not independent. C. Given C = {HHH, TTT}, A = {HHH}, B = {HHH}, [tex]A\cap B = {HHH}[/tex], i.e., P(A|C)=1/2, P(B|C)=1/2 and [tex]P(A\cap B|C)=1/2[/tex]. Because [tex]P(A\cap B|C) = 1/2\neq (1/2)(1/2) = P(A|C)P(B|C)[/tex] events A and B are not conditionally independent given C. D. Given B = {THH, HHH}, A = {HHH}, C={HHH}, [tex]A\cap C = {HHH}[/tex], i.e., P(A|B)=1/2, P(C|B)=1/2 and [tex]P(A\cap C|B)=1/2[/tex]. Because [tex]P(A\cap C|B) = 1/2\neq (1/2)(1/2) = P(A|B)P(C|B)[/tex] events A and C are not conditionally independent given B. E. The probability of winning is P(C) = 2/8 = 1/4