Assertion (A) : X is a binomial distribution with parameters n = 100 and p .
If P (x = 50) = P ( x = 49) then p = `(1)/(2)`
Reason (R) : For the binomial distribution `(q + p)^n , P (x = k) = ""^(n) c_(k) .q^(n-k) .p^(k)`

Options

A. A is false , R is true
B. A is true , R is fals
C. A , R are true but R `implies` A
D. A , R are true , and `R cancelimplies A`

Answer

To solve the problem, we need to analyze the assertion and the reason given in the question.

### Step 1: Understand the Assertion
The assertion states that if \( P(X = 50) = P(X = 49) \) for a binomial distribution with parameters \( n = 100 \) and \( p \), then \( p = \frac{1}{2} \).

### Step 2: Write the Probability Formulas
For a binomial distribution, the probability mass function is given by:
\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]
where \( q = 1 - p \).

### Step 3: Set Up the Equations
We need to set up the equations for \( P(X = 50) \) and \( P(X = 49) \):
1. \( P(X = 50) = \binom{100}{50} p^{50} q^{50} \)
2. \( P(X = 49) = \binom{100}{49} p^{49} q^{51} \)

### Step 4: Equate the Two Probabilities
Set the two probabilities equal to each other:
\[ \binom{100}{50} p^{50} q^{50} = \binom{100}{49} p^{49} q^{51} \]

### Step 5: Simplify the Equation
Using the property of binomial coefficients:
\[ \binom{100}{49} = \frac{100}{51} \binom{100}{50} \]
Substituting this into the equation gives:
\[ \binom{100}{50} p^{50} q^{50} = \frac{100}{51} \binom{100}{50} p^{49} q^{51} \]
Dividing both sides by \( \binom{100}{50} \) (assuming it is not zero):
\[ p^{50} q^{50} = \frac{100}{51} p^{49} q^{51} \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ p q = \frac{100}{51} \]

### Step 7: Substitute \( q \)
Since \( q = 1 - p \), we can substitute:
\[ p(1 - p) = \frac{100}{51} \]

### Step 8: Solve the Quadratic Equation
This leads to the quadratic equation:
\[ p - p^2 = \frac{100}{51} \]
Rearranging gives:
\[ p^2 - p + \frac{100}{51} = 0 \]

### Step 9: Use the Quadratic Formula
Using the quadratic formula \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ p = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot \frac{100}{51}}}{2} \]

### Step 10: Analyze the Roots
Calculating the discriminant:
\[ 1 - \frac{400}{51} = \frac{51 - 400}{51} = \frac{-349}{51} \]
Since the discriminant is negative, there are no real solutions for \( p \) that satisfy the equation.

### Conclusion
Thus, the assertion \( P(X = 50) = P(X = 49) \) does not imply \( p = \frac{1}{2} \), making the assertion false. The reason provided is true as it correctly describes the binomial distribution.

### Final Answer
- Assertion (A) is false.
- Reason (R) is true.