Question

Let X denote the number of bits received in error in a digital communication channel, and assume that X is a binomial with p = 0.0005. If 1000 bits are transmitted, determine the following Round your answers to four decimal places (e.g. 98.7654)

P(X= 1)

Answer 1

To determine the probability P(X=1) where X follows a binomial distribution with parameters n=1000 and p=0.0005, insert the values into the binomial probability formula. Calculate the binomial coefficient C(1000,1), raise p and (1-p) to their respective powers, and multiply the terms to find the probability to four decimal places.

Step-by-step explanation:

The question involves the binomial distribution where X is the number of bits received in error. Given that X is a binomial random variable with parameters n = 1000 and p = 0.0005, we are interested in finding P(X=1), which is the probability that exactly one bit is received in error out of the 1000 bits transmitted.

To calculate this, we use the binomial probability formula:
P(X=k) = C(n, k) * p^k * (1-p)^(n-k),
where k is the number of successes, n is the number of trials, C(n, k) is the binomial coefficient, and p is the probability of success on a single trial.

To find P(X=1), substitute the values into the formula:
P(X=1) = C(1000, 1) * 0.0005^1 * (1-0.0005)^(1000-1).
This will give you the requested probability rounded to four decimal places.