A memoryless binary symmetric channel has bit error probability $10^{-5}$. If $10^5$ bits are sent, the probability that no more than one bit is wrong is ________.
Let N be the error count. For n=100000 and p=$10^{-5}$, it has a binomial distribution. Therefore $$P(N\le1)=(1-p)^n+np(1-p)^{n-1}.$$ Since np=1, the Poisson approximation gives $e^{-1}(1+1)=2/e=0.735759$. The exact binomial value is essentially the same, **0.73576**, within the official 0.70–0.75 range.
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