Final answer:
The probability that a 13-bit message is received with at most one error is calculated by adding the binomial probabilities of receiving the message with exactly zero errors and with exactly one error, using the given success rate of 87% for each bit.
Step-by-step explanation:
To calculate the probability that a 13-bit message transmitted through a channel is received with at most one error, we apply the principles of binomial probability. Given that each bit has an 87% chance of being relayed correctly, the probability of getting exactly zero errors (all bits correct) is found using the binomial probability formula:
P(X = 0) = (13 choose 0) * (0.87)^13 * (0.13)^0
This formula represents the probability of selecting 0 bits to be in error from the 13 bits, multiplied by the probability that 13 bits are transmitted correctly, raised to the power of their respective occurrences (with 0.87 being the success rate and 0.13 being the failure rate).
Similarly, the probability of getting exactly one error is:
P(X = 1) = (13 choose 1) * (0.87)^12 * (0.13)^1
To find the probability of at most one error, we sum these two probabilities:
P(X ≤ 1) = P(X = 0) + P(X = 1)
After calculating the values using the binomial coefficients and the given probabilities, we add them to get the final probability.
The subject of this calculation falls within the area of arithmetic and probability theory, both of which are integral parts of high school level mathematics.