The following questions consist of two statements, one labelled as the 'Assertion (A)' and the other as 'Reason (R)'. You are to examine these two statements carefully and select the answer.
Assertion (A) : For a binomial distribution B(n, p), Mean gt Variance

Reason (R) : Probability is less than or equal to 1

Options

A. Both A and R individually true and R is the correct explanation of A
B. Both A and R are individually true but R is not the correct explanation of A.
C. A is true but R is false.
D. A is false but R is true.

Explanation

To solve the problem, we need to analyze the two statements provided: the Assertion (A) and the Reason (R).

### Step 1: Understand the Assertion (A)
The assertion states that for a binomial distribution \( B(n, p) \), the mean is greater than the variance.

- The mean \( \mu \) of a binomial distribution is given by:
\[ \mu = np \]
- The variance \( \sigma^2 \) of a binomial distribution is given by:
\[ \sigma^2 = np(1 - p) \]

### Step 2: Compare Mean and Variance
We need to check if \( np > np(1 - p) \).

1. Start by simplifying the inequality:
\[ np > np(1 - p) \]
Dividing both sides by \( np \) (assuming \( np \neq 0 \)):
\[ 1 > 1 - p \]
This simplifies to:
\[ p > 0 \]
Since \( p \) is a probability, it is always between 0 and 1 (i.e., \( 0 < p \leq 1 \)). Therefore, the assertion \( np > np(1 - p) \) holds true for \( 0 < p < 1 \).

### Step 3: Understand the Reason (R)
The reason states that probability is less than or equal to 1.

This statement is true because probabilities are defined to be within the range of 0 to 1, inclusive. However, this does not directly explain why the mean is greater than the variance in a binomial distribution.

### Step 4: Conclusion
Both the assertion and the reason are true statements:
- The assertion is true because the mean of a binomial distribution is indeed greater than its variance for valid values of \( p \).
- The reason is true, but it does not provide a valid explanation for the assertion.

### Final Answer
The correct option is: **Both assertion and reason are individually true, but the reason is not the correct explanation of the assertion.**

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