Question

Suppose a 5-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is 0.01. Flipping of each bit is independent of one another. The probability that the message is delivered error-free to the destination is ___________. (rounded off to three decimal places)

Hint:

When calculating the probability that all bits in a message are transmitted error-free, multiply the probability of each bit being error-free, assuming the bits are independent.

Answer 1

Correct Answer: 0.949

A transmission error occurs only if at least one of the five bits flips during transmission. Therefore, it is easier to first consider the opposite event: the situation in which no bit flips at all.

Each bit independently remains unchanged with probability \(0.99\). For the entire 5-bit message to arrive without any error, every bit must remain unchanged simultaneously.

The likelihood of this happening is:

\(0.99 \times 0.99 \times 0.99 \times 0.99 \times 0.99 = 0.99^5\)

Evaluating this expression gives:

\(0.99^5 \approx 0.95099\)

Hence, the probability that the message is delivered without any error is \(\boxed{0.951}\) (rounded to three decimal places).