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)
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Correct Answer: 0.949
Solution and Explanation
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).
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