For `X ~ B(n , P)` and `P(X=x) = ""^(8)C_(x) (1//2)^(x) (1//2)^(8-x)` then state values of n and P

Answer

To solve the problem, we need to identify the parameters \( n \) and \( P \) from the given probability mass function of a binomial distribution \( P(X=x) \).

### Step-by-Step Solution:

1. **Identify the General Form of the Binomial Distribution:**
The probability mass function for a binomial distribution is given by:
\[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \]
where:
- \( n \) is the number of trials,
- \( p \) is the probability of success,
- \( x \) is the number of successes.

2. **Compare with the Given Expression:**
The given expression is:
\[ P(X = x) = \binom{8}{x} \left(\frac{1}{2}\right)^x \left(\frac{1}{2}\right)^{8-x} \]
We can rewrite this as:
\[ P(X = x) = \binom{8}{x} \left(\frac{1}{2}\right)^8 \]

3. **Identify \( n \):**
From the expression, we can see that the coefficient \( \binom{8}{x} \) indicates that \( n = 8 \).

4. **Identify \( p \):**
The term \( \left(\frac{1}{2}\right)^x \) corresponds to \( p^x \), which implies that \( p = \frac{1}{2} \).

5. **Determine \( q \):**
Since \( q = 1 - p \), we have:
\[ q = 1 - \frac{1}{2} = \frac{1}{2} \]

### Conclusion:
Thus, the values of \( n \) and \( P \) are:
- \( n = 8 \)
- \( P = \frac{1}{2} \)