For a binomial variate X , if `n = 4 and P(X=0) = 16/81` then `P(X =4)` is

Options

A. `1/3`
B. `1/27`
C. `1/81`
D. `1/16`

Answer

To solve the problem step by step, we need to find \( P(X = 4) \) given that \( n = 4 \) and \( P(X = 0) = \frac{16}{81} \).

### Step 1: Understand the Binomial Probability Formula
The probability of getting exactly \( x \) successes in \( n \) trials in a binomial distribution is given by the formula:
\[ P(X = x) = \binom{n}{x} p^x q^{n-x} \]
where:
- \( n \) is the number of trials,
- \( x \) is the number of successes,
- \( p \) is the probability of success,
- \( q = 1 - p \) is the probability of failure.

### Step 2: Apply the Formula for \( P(X = 0) \)
We know that \( P(X = 0) = \frac{16}{81} \). Using the formula:
\[ P(X = 0) = \binom{4}{0} p^0 q^4 \]
Since \( \binom{4}{0} = 1 \) and \( p^0 = 1 \), we have:
\[ P(X = 0) = q^4 = \frac{16}{81} \]

### Step 3: Solve for \( q \)
From the equation \( q^4 = \frac{16}{81} \), we can take the fourth root of both sides:
\[ q = \left(\frac{16}{81}\right)^{1/4} \]
Calculating this gives:
\[ q = \frac{2}{3} \]

### Step 4: Find \( p \)
Since \( p + q = 1 \), we can find \( p \):
\[ p = 1 - q = 1 - \frac{2}{3} = \frac{1}{3} \]

### Step 5: Calculate \( P(X = 4) \)
Now we need to find \( P(X = 4) \):
\[ P(X = 4) = \binom{4}{4} p^4 q^{4-4} \]
Again, since \( \binom{4}{4} = 1 \) and \( q^0 = 1 \), we have:
\[ P(X = 4) = p^4 = \left(\frac{1}{3}\right)^4 \]
Calculating this gives:
\[ P(X = 4) = \frac{1}{81} \]

### Final Answer
Thus, the probability \( P(X = 4) \) is:
\[ \boxed{\frac{1}{81}} \]