Binomial Distribution is a probability distribution that represents the number of successes in a fixed number of independent Bernoulli trials. It is characterized by two parameters: the probability of success (p) and the probability of failure (q = 1 - p).

Symmetry in Binomial Distribution:
Symmetry refers to a distribution that is balanced or equal on both sides. In the context of binomial distribution, symmetry occurs when the probabilities of obtaining a certain number of successes on one side of the distribution are equal to the probabilities of obtaining the same number of successes on the other side.

The correct answer to the given question is option 'C' that states "p = q". This implies that the probability of success (p) is equal to the probability of failure (q).

Explanation:

1. Definition of Symmetry:
Symmetry in statistics refers to a balanced distribution where the probabilities or values on one side of the center are the mirror image of the probabilities or values on the other side. In other words, the distribution is identical on both sides.

2. Binomial Distribution:
Binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success (p).

3. Parameters of Binomial Distribution:
The binomial distribution is characterized by two parameters:
- The probability of success in each trial (p)
- The probability of failure in each trial (q = 1 - p)

4. Symmetry in Binomial Distribution:
In a binomial distribution, the probabilities of obtaining a certain number of successes on one side of the distribution should be equal to the probabilities of obtaining the same number of successes on the other side.

5. p = q:
If p = q, it means that the probability of success (p) is equal to the probability of failure (q). This condition satisfies the requirement for symmetry in the binomial distribution.

6. Example:
Let's consider an example where p = 0.5 and q = 0.5. This means that the probability of success and failure in each trial is equal. In this case, the probabilities of obtaining 0, 1, 2, 3, etc. successes will be symmetrical on both sides of the distribution. For example, the probability of obtaining 0 successes is the same as the probability of obtaining all failures, and the probability of obtaining 1 success is the same as the probability of obtaining 1 failure.

Therefore, when p = q, the binomial distribution is symmetrical.