Question

In a digital communication channel, assume that the number of bits received in error can be modeled by a binomial random variable. The probability that a bit is received in error is 0.1. A) If 50 bits are transmitted, what is the probability that 2 or fewer errors occur? ( Round your answer to 3 decimal places) B) If 50 bits are transmitted, what is the probability that more than 8 errors occur? ( Round your answer to 5 decimal places)

Answer 1

a)
Here, n = 50, p = 0.1, (1 - p) = 0.9 and x = 2
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X <= 2).
P(X <= 2) = (50C0 * 0.1^0 * 0.9^50) + (50C1 * 0.1^1 * 0.9^49) + (50C2 * 0.1^2 * 0.9^48)
P(X <= 2) = 0.005 + 0.029 + 0.078
P(X <= 2) = 0.112

b)
P(X <= 8) = (50C0 * 0.1^0 * 0.9^50) + (50C1 * 0.1^1 * 0.9^49) + (50C2 * 0.1^2 * 0.9^48) + (50C3 * 0.1^3 * 0.9^47) + (50C4 * 0.1^4 * 0.9^46) + (50C5 * 0.1^5 * 0.9^45) + (50C6 * 0.1^6 * 0.9^44) + (50C7 * 0.1^7 * 0.9^43) + (50C8 * 0.1^8 * 0.9^42)
P(X <= 8) = 0.00515 + 0.02863 + 0.07794 + 0.13857 + 0.1809 + 0.18492 + 0.1541 + 0.10763 + 0.06428
P(X <= 8) = 0.94212

P(X > 8) = 1 - 0.94212 = 0.05788