GATE EC 2020 Set 1 — Question 22
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Communications → Digital Modulation & Detection → Binomial Distribution
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Question
A digital communication system transmits a block of bits. The probability of error in decoding a bit is . The error event of each bit is independent of the error events of the other bits. The received block is declared erroneous if at least one of its bits is decoded wrongly. The probability that the received block is erroneous is
Correct answer
(D) 1 - (1-α)^N
Solution
Let be the probability of error in decoding a single bit. Given .
The probability of a bit being decoded correctly is .The system transmits a block of bits. The error events for each bit are independent.The received block is declared erroneous if at least one of its bits is decoded wrongly.
It is easier to calculate the probability of the complementary event: the block is not erroneous.The block is not erroneous if and only if all bits are decoded correctly.
Since the error events are independent, the probability that all bits are decoded correctly is the product of the probabilities of each bit being correct:
The probability that the received block is erroneous is minus the probability that all bits are correct:
This matches option (D).The final answer is \boxed{\text{1 - (1-\alpha)^N}}
The probability of a bit being decoded correctly is .The system transmits a block of bits. The error events for each bit are independent.The received block is declared erroneous if at least one of its bits is decoded wrongly.
It is easier to calculate the probability of the complementary event: the block is not erroneous.The block is not erroneous if and only if all bits are decoded correctly.
Since the error events are independent, the probability that all bits are decoded correctly is the product of the probabilities of each bit being correct:
The probability that the received block is erroneous is minus the probability that all bits are correct:
This matches option (D).The final answer is \boxed{\text{1 - (1-\alpha)^N}}
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