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

A. TRUE B. FALSE C. FALSE D. FALSE E. FALSE

Step-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 , 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 , we have that A and C are not independent.

C. Given C = {HHH, TTT}, A = {HHH}, B = {HHH}, , i.e., P(A|C)=1/2, P(B|C)=1/2 and . Because events A and B are not conditionally independent given C.

D. Given B = {THH, HHH}, A = {HHH}, C={HHH}, , i.e., P(A|B)=1/2, P(C|B)=1/2 and . Because events A and C are not conditionally independent given B.

E. The probability of winning is P(C) = 2/8 = 1/4