One of two equally likely messages is sent across a communication channel to a r

Question

One of two equally likely messages is sent across a communication channel to a receiver. The first message is AAA and the second message is BBB. The probability of any letter being received in error (e.g., A received as B, or vice versa) is 0.1, independent of other letters in the word. Suppose that the received message is ABA. What is the probability that the transmitted message was BBB? Consider a random access communication channel employed by multiple users. If more than one user attempts to use the channel during a given timeslot, there is a "collision," and the timeslot is wasted. Assume that each user attempts to use a specific timeslot (independent of other users) with probability p. If there are n users, what is the probability of a collision?

Explanation / Answer

Question 10:

Here we are given that both the messages are equally likely, therefore,

P(AAA) = P(BBB) = 0.5 as both the messages are equally likely.

Now finding the probability that ABA message is received given that AAA messages was sent as:

P(ABA | AAA) = 0.9*0.1*0.9 = 0.081

Now finding the probability that ABA message is received given that BBB messages was sent as:

P(ABA | BBB) = 0.1*0.9*0.1 = 0.009

Therefore the probability that the message ABA is received would be computed as:

P(ABA) = P(ABA | AAA)*P(AAA) + P(ABA | BBB)*P(BBB) = 0.5*0.081 + 0.5*0.009 = 0.045

Now using the Bayes theorem we get:

P(BBB | ABA)P(ABA) = P(ABA | BBB)*P(BBB)

Putting all the given values in the above equation we get:

P(BBB | ABA)*0.045 = 0.5*0.009

P(BBB | ABA) = 0.5*0.009 / 0.045 = 0.1

Therefore 0.1 is the required probability here that given the message received is ABA, the message sent was BBB.

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