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Homework had this question. Part a is relatively straightforward, as it's a cdf of the function. For part b I could write a script that will compute all possible sets and divide by the total sets, but how would I solve this problem in an efficient way?

A binary communication channel transmits a sequence of "bits" (0s and 1s). Suppose that for any particular bit transmitted, there is a 15% chance of a transmission error (a 0 becoming a 1 or a 1 becoming a 0). Assume that bit errors occur independently of one another. (Round your answers to four decimal places.)

(a) Consider transmitting 1000 bits. What is the approximate probability that at most 175 transmission errors occur?

.9868

Correct: Your answer is correct.

(b) Suppose the same 1000-bit message is sent two different times independently of one another. What is the approximate probability that the number of errors in the first transmission is within 60 of the number of errors in the second?

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  • $\begingroup$ The key hint in the question is the word approximate. Use the normal approximation of the binomial distribution. $\endgroup$ Nov 27, 2019 at 0:38
  • $\begingroup$ Let X be the number of errors in transmission. It follows a.Binomial distribution. For part a) you should have used the Central Limit Theorem to estimate the distribution of X as a Normal distribution N(u, var). Now for part b), you have two independent normal random variables X, Y ~ N(u, var) and you want to compute P(|X-Y| < 60). Note that X-Y is a normal distribution with mean 0 and variance 2*var. So with some algebra you can express the desired probability as the probability that a standard normal variable lies some number of std deviations from the mean, which you can look up in a table. $\endgroup$ Nov 27, 2019 at 0:39
  • $\begingroup$ So I used std deviation 255 for X-Y but it says my answer is wrong? What can I do to fix it $\endgroup$
    – pasha
    Nov 27, 2019 at 1:49
  • $\begingroup$ @pasha The variance should be 255, so the standard deviation should be about 16. $\endgroup$ Nov 27, 2019 at 2:21

1 Answer 1

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As answered by @Ragib Zaman, I have to use the central limit theorem to calculate the answer instead of treating it as a binomial answer).

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