Laboratory Project 2 Project on Conditional Probabilities Communication through a Noisy Channel
- Probability of erroneous transmission Introduction: Send a one-bit message S and check the received signal R. if R and S are the same then consider the experiment success. Otherwise it is a failure. The given probabilities are P0 = 0.4, E0 = 0.02 and E1 = 0.03 Methodology: The methodology I used is generate a random number m = random.uniform() and generate the transmitted message S as when the m > P0 then S == 1 or m <= P0 then S == 0. On receive signal R, generate a random number t = random.uniform() and different than m. The relationship table will be: S=0 If m<= P0 S=1 If m>P0
R =1 If S =0 T<=E0 R =0 If S =0 T>E0 R =1 If S =1 T>E1 R =0 If S =1 T<=E1
Result: get the probability of transmission error with 1-P(success). Probability of transmission error Ans. P = 0.02515
- Conditional probability: P(R=1|S=1)
Introduction: This experiment is send a one-bit message S but the conditional of S will be set as S=1. Since the S is always 1, when received R == 1 then called the experiment is a success otherwise failure. The required probabilities E1 is give as 0.03 Methodology: The relationship table I used in this experiment is as below since S is always 1 R =1 If S =1 T>E1 R =0 If S =1 T<=E1
Conclusion: get the probability of success using number of successes divide 100000 experiment times. Result: Conditional probability P(R=1|S=1) Ans. P = 0.97033
- Conditional probability: P(S=1|R=1)
Introduction: This experiment is received a one-bit message R but the conditional of R will be set as R=1. Since the R is always 1, when send S == 1 then called the experiment is a success otherwise failure. The required probabilities P0, E0 and E1 are given as 0.4, 0.02 and 0.03. Methodology: The relationship table I used in this experiment is as below since R is always 1 S=1 If m>P0 And the relationship table of R =1 If S =0 T<=E0 R =1 If S =1 T>E1
Conclusion: get the probability of success using number of successes divide total count of when message S ==1. Result: Conditional probability P(S=1|R=1) Ans. p= 0.982134
- Enhanced transmission method
Introduction: This experiment is create and send a one-bit message S three times. And the receive side R will be set as 000, 001, 010, 100, 011, 101, 110, 111. When checking the transmission is success use majority rule.
Methodology: To apply majority rule, use table below as look up table
When R = 000 , 001, 010 , 100 D = 0
When R = 011, 101, 110, 111 D = 1
D( R ) == D ( S ), such as the sum of S is same as the sum of R The experiment is success
Result:
Get the probability of transmission incorrectly with 1-P(success).
Probability of error with enhanced transmission Ans. p= 0.00204
Comment:
Using this voting method dramatically decreased the probability of error compare method than Problem 1.