When a message is sent electronically, it is usually sent as a stream of bits, each of which can be either a 0 or 1. If the digital channel is noisy, then each bit has some probability of being flipped.
Assume that a message is sent through a noisy channel where the probability that any individual bit will be flipped is 0.2. What is the probability that a message is 4 bits long will be successfully transmitted?
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For the answers below, P(F) = P(a bit is flipped) = 0.2.
Therefore, P(S) = P(a bit is successfully transmitted) = 1 - P(F) = 0.8.
1. Probability that a message 4 bits long is successfully transmitted
= P(S) ^ 4
= 0.8 ^ 4
2. Probability that a 7-bit message gets transmitted w/ no more than 1 bit being flipped (note 7C1 = 7 combination 1 = 7)
= P(0 bits flipped) + P(1 bit flipped)
= P(S)^7 + 7C1 * P(S)^6 * P(F)
= 0.8^7 + 7 * 0.8^6 * 0.2
= 0.2097152 + 0.3670016
3. As stated in the problem, the message ...
The solution breaks down the problem into 4 manageable parts with details on how to solve each.