Question

In a communication system, each codeword consists of 1000 bits. Due to the noise, each bit may be received in error with probability 0.1. It is assumed bit errors occur independently. Since error correcting codes are used in this system, each codeword can be decoded reliably if there are less than or equal to 125 errors in the received codeword, otherwise the decoding fails. Using the CLT, find the probability of decoding failure?

Answer 1

To find the probability of decoding failure, we can use the Central Limit Theorem (CLT). By calculating the mean and standard deviation of the number of errors in a codeword, we can determine the probability of decoding failure. Plugging in the values, we find that the probability of decoding failure is approximately 0.004 or 0.4%.

Step-by-step explanation:

To use the Central Limit Theorem (CLT) to find the probability of decoding failure, we need to calculate the mean and standard deviation of the number of errors in a codeword. The mean number of errors is the product of the probability of error (0.1) and the number of bits (1000), which is 100. The standard deviation is the square root of the product of the probability of success (0.9) and the number of bits, which is 9.487.

Next, we calculate the z-score using the formula:

z = (x - mean) / standard deviation

where x is the number of errors that lead to decoding failure (126). Plugging in the values, we get:

z = (126 - 100) / 9.487 = 2.742

Using a standard normal distribution table or a calculator, we can find the probability associated with a z-score of 2.742 to be approximately 0.996. Therefore, the probability of decoding failure is approximately 0.004 or 0.4%.