What is the probability of exactly one error if 200 bits are transmitted through a binary symmet

What is the probability of exactly one error if 200 bits are...

Key Concepts

Probability Mass Function

The Probability Mass Function (PMF) specifies the probability that a discrete random variable equals a particular value. For the binomial setting, it combines both the number of ways to achieve exactly a specific number of errors (through combinatorial coefficients) and the probabilities associated with each error and correct transmission, thus providing the exact likelihood of encountering a designated number of errors in the transmitted data.

Binary Symmetric Channel

A Binary Symmetric Channel (BSC) is a simplified mathematical model used in digital communications where each transmitted bit has an equal probability of being flipped during transmission. This channel model assumes that errors occur independently and uniformly, meaning the chance of a bit flipping from 0 to 1 is the same as from 1 to 0, making it a fundamental basis for analyzing error behavior in binary data transmission systems.

Binomial Distribution

The Binomial Distribution is a probability distribution that models the number of successes or failures in a fixed number of independent Bernoulli trials with the same success probability. In the context of transmission systems, it is used to calculate the probability of obtaining exactly a certain number of errors out of a given number of transmitted bits, reflecting the random and independent nature of each bit's potential error.

Transcript

00:02 We're asked questions about binary data transmitted over a noisy communication channel.

00:08 We're told that the probability that a received binary digit is an error due to channel noise is 0 .05.

00:17 And we're told to assume that such errors occur independently within the bit stream.

00:22 In part a, we're asked to find the probability that the third error occurs on the 50th transmitted bit.

00:50 First, let's define some variables.

00:54 So we'll let x be the number of bits transmitted until the third error occurs.

01:24 Then we have that x is distributed according to a negative binomial distribution with parameters r equals 3.

01:46 This is the number of failures and the probability of failure p equals 0 .05.

01:54 So it follows that the probability of the number of bits transmitted until the third error occurs is 50, or that x is equal to 50.

02:12 This is by the formula for the negative binomial.

02:18 This is nb of 50 with parameters 3 and 0 .05, which is going to be 50 minus 1, choose 3 minus 1 times probability of failure, 0 .05 to the third power for each failure, that's the 3, times the probability of success .95 to the number of successes, which is 50 minus 3 or 47.

03:16 Using a calculator, we find that this is approximately .013.

03:35 In part b, we're asked to determine on average how many bits will be transmitted correctly before the first error.

03:45 So this is not including the first error.

03:49 Well, we use the mean of the geometric distribution.

03:56 So the average number of bits up to an including the first error.

04:25 Since this is a geometric distribution, the mean is 1 over p, which is 1 over 0 .05, or in other words, 20 bits...

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