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

We are given the following information:
- The message is 5 bits long.
- The probability that a bit gets flipped during transmission is \( 0.01 \).
- The flipping of each bit is independent of one another.

Step 1: Probability of a bit being transmitted error-free

The probability that a given bit is transmitted error-free (i.e., it is not flipped) is the complement of the probability that the bit is flipped:
\[ P(\text{bit is error-free}) = 1 - 0.01 = 0.99. \] Step 2: Probability that all 5 bits are transmitted error-free

Since the flipping of each bit is independent, the probability that all 5 bits are transmitted error-free is the product of the probabilities for each bit:
\[ P(\text{message delivered error-free}) = (0.99)^5. \] Step 3: Calculate the value

Now, calculate \( (0.99)^5 \):
\[ (0.99)^5 \approx 0.95099. \] Final Answer:
Thus, the probability that the message is delivered error-free to the destination is \( \boxed{0.951} \) (rounded to three decimal places).