GATE2025_CS1_Q56

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)

✅ Final Answer:

The correct answer is in the range 0.949 to 0.952. The exact value is approximately 0.951.

🔹 Short Answer:

For the message to be error-free, all 5 bits must be transmitted correctly. The probability of one bit being transmitted correctly is 1 - 0.01 = 0.99. Since the events are independent, the probability of all 5 bits being correct is (0.99)⁵. This calculates to approximately 0.95099, which rounds to 0.951.

🔸 Long Answer:

This problem deals with the probability of multiple independent events occurring in a sequence. Let's break down the logic.

Step 1: Define the Probabilities of a Single Event

We are given the probability of a single bit getting flipped (an error).

• Probability of a bit being flipped (Error), P(E) = 0.01

The complementary event is that a bit is transmitted correctly (no error). The probability of a single bit being correct is:

P(Correct Bit) = 1 - P(E)
P(Correct Bit) = 1 - 0.01 = 0.99

Step 2: Define the Condition for an Error-Free Message

An "error-free" message means that every single bit in the 5-bit message must be transmitted correctly. This can be expressed as:

P(Error-Free Message) = P(1st bit is correct AND 2nd bit is correct AND 3rd bit is correct AND 4th bit is correct AND 5th bit is correct)

Step 3: Apply the Rule of Independent Events

The problem states that "Flipping of each bit is independent of one another." This is a key piece of information. For independent events, the probability of all of them happening is the product of their individual probabilities.

P(A and B) = P(A) × P(B)    (for independent A, B)

Applying this to our 5-bit message:

P(Error-Free Message) = P(Correct Bit) × P(Correct Bit) × P(Correct Bit) × P(Correct Bit) × P(Correct Bit)
= [P(Correct Bit)]⁵

Step 4: Calculate the Final Probability

Substitute the value we found in Step 1:

P(Error-Free Message) = (0.99)⁵
= 0.99 × 0.99 × 0.99 × 0.99 × 0.99
= 0.9509900499

Step 5: Format the Answer

The question asks for the answer to be rounded off to three decimal places.

• The first three decimal places are 950.
• The fourth decimal place is 9, which is 5 or greater, so we round up the third decimal place.

0.95099... ≈ 0.951

This value falls within the accepted GATE answer range of 0.949 to 0.952.

Bibliography:
- Sheldon, R. (2014). A First Course in Probability. Pearson. (Chapter 3: Conditional Probability and Independence).
- Spiegel, M. R., Schiller, J. J., & Srinivasan, R. A. (2013). Schaum's Outline of Probability and Statistics. McGraw-Hill Education.

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