**Binary Symmetric Channel (BSC):**
A binary symmetric channel (BSC) is a communication channel with two possible input symbols (0 and 1) and two possible output symbols (also 0 and 1). The channel introduces errors in transmission, where the transmitted symbol may be flipped with a certain probability. The transition probability, denoted by p, represents the probability of an error occurring during transmission.

**Probability of Error for Optimum Receiver:**
To determine the probability of error for an optimum receiver in a BSC, we need to consider the transition probability and the probabilities of the transmitted symbols.

Given:
Transition probability, p = 1/8
Probability of symbol 0, P(X = 0) = 9/10

**Calculating the Probability of Error:**
The probability of error can be calculated using the following formula:

P(error) = P(0|1) * P(1) + P(1|0) * P(0)

P(0|1) represents the probability of receiving a 0 when a 1 was transmitted, and P(1|0) represents the probability of receiving a 1 when a 0 was transmitted.

In a BSC, the transition probability is the same for both error types, so P(0|1) = P(1|0) = p.
Substituting the given values:

P(error) = p * P(1) + p * P(0)

P(1) represents the probability of transmitting a 1, and P(0) represents the probability of transmitting a 0.

Since the channel is binary, P(0) + P(1) = 1.

P(error) = p * P(1) + p * (1 - P(1))

P(error) = p

Therefore, the probability of error for an optimum receiver in this BSC is equal to the transition probability, which is 1/8.

**Explanation:**
In the given scenario, the transition probability of the BSC is 1/8. This means that there is a 1/8 chance that an error will occur during transmission. The probability of symbol 0 being transmitted is 9/10.

To calculate the probability of error for an optimum receiver, we use the formula P(error) = p * P(1) + p * (1 - P(1)), where p is the transition probability, P(1) is the probability of transmitting a 1, and P(0) = 1 - P(1).

Substituting the values, we get P(error) = (1/8) * (1) + (1/8) * (1 - 1/10) = 1/8.

Therefore, the probability of error for an optimum receiver in this BSC is 1/8, which is the same as the transition probability. This means that there is a 1/8 chance of receiving an incorrect symbol at the receiver's end.

Binary symmetric channel is like probability of receiving zero when one is transmitted and receiving one when zero is transmitted are equal they gave it 1/8 so error probability is 2/8.