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Packet Transmission

A packet consisting of a string of symbols is transmitted over a noisy channel. Each symbol has a probability 0.0001 of being transmitted in error. Find the largest for which the probability of incorrect transmission (at least one symbol in error) is less than 0.001.

Wolfram Language code: transmissionProcess = BinomialProcess[10 ^ -4];
The probability of an error in the transmission:
Wolfram Language code: prob = Probability[x > 0, xtransmissionProcess[n]]
Wolfram Language code: ListLogPlot[Transpose@Table[{prob, 10 ^ (-3)}, {n, 0, 15}], Joined -> True, Filling -> Axis]
Find the largest for which the probability of incorrect transmission is less than :
Wolfram Language code: Maximize[{n, prob <= 10 ^ -3 && n∈Integers && 5 <= n <= 15}, n]