6. A ternary string is a string consisting of Os, 1s, and 2s. Suppose we are tra

Question

6. A ternary string is a string consisting of Os, 1s, and 2s. Suppose we are transmitting ternary strings through a noisy communication channel with the following known probabilities: • 0 is sent with probability 0.3 • 1 is sent with probability 0.4 2 is sent with probability 0.3 Due to noise, 0 is changed to 1 during transmission with probability 0.2 • Due to noise, 0 is changed to 2 during transmission with probability 0.1 Due to noise, 1 is changed to 0 during transmission with probability 0.2 Due to noise, 1 is changed to 2 during transmission with probability 0.1 Due to noise, 2 is changed to 0 during transmission with probability 0.2 • Due to noise, 2 is changed to 1 during transmission with probability 0.1 Suppose that a l is received. What is the probability that a l was sent? You must use both Bayes' Formula and the theorem of Total Probability to answer this question. Show all of your work.

Explanation / Answer

from total probability:

probability that 1 is received =P( 0 is sent and changed to 1+1 is sent and changed to 1+2 is sent and changed to 1)

=0.3*0.2+0.4*(1-0.2-0.1)+0.3*0.1=0.37

therefore from Bayes theorum: probability that a 1 was sent given 1 is received

=P(1 is sent and changed to 1)/P(1 is received) =0.4*(1-0.2-0.1)/0.37=0.7568

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