Let a random variable X have a binomial distribution with mean 8 and variance 4. If `P(x le2)=(k)/(2^(16))`, then `(k-47)/(10)` is equal to __________.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the properties of the binomial distribution
The mean (μ) and variance (σ²) of a binomial distribution with parameters n (number of trials) and p (probability of success) are given by:
- Mean: \( \mu = np \)
- Variance: \( \sigma^2 = npq \) where \( q = 1 - p \)

### Step 2: Set up the equations based on the given mean and variance
From the problem, we know:
- Mean \( np = 8 \)
- Variance \( npq = 4 \)

### Step 3: Express q in terms of p
Since \( q = 1 - p \), we can substitute this into the variance equation:
\[ np(1 - p) = 4 \]

### Step 4: Substitute np from the mean into the variance equation
From the mean equation, we have \( n = \frac{8}{p} \). Substituting this into the variance equation gives:
\[ \frac{8}{p} \cdot p \cdot (1 - p) = 4 \]
This simplifies to:
\[ 8(1 - p) = 4 \]
\[ 8 - 8p = 4 \]
\[ 8p = 4 \]
\[ p = \frac{1}{2} \]

### Step 5: Find q
Now, since \( q = 1 - p \):
\[ q = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 6: Substitute p back to find n
Using \( np = 8 \):
\[ n \cdot \frac{1}{2} = 8 \]
\[ n = 16 \]

### Step 7: Calculate \( P(X \leq 2) \)
We need to find \( P(X \leq 2) \):
\[ P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) \]
Using the binomial probability formula:
\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]

Calculating each term:
1. \( P(X = 0) = \binom{16}{0} \left(\frac{1}{2}\right)^0 \left(\frac{1}{2}\right)^{16} = 1 \cdot 1 \cdot \frac{1}{65536} = \frac{1}{65536} \)
2. \( P(X = 1) = \binom{16}{1} \left(\frac{1}{2}\right)^1 \left(\frac{1}{2}\right)^{15} = 16 \cdot \frac{1}{2} \cdot \frac{1}{32768} = \frac{16}{65536} \)
3. \( P(X = 2) = \binom{16}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^{14} = 120 \cdot \frac{1}{4} \cdot \frac{1}{16384} = \frac{120}{65536} \)

### Step 8: Sum the probabilities
Now, we sum these probabilities:
\[ P(X \leq 2) = \frac{1 + 16 + 120}{65536} = \frac{137}{65536} \]

### Step 9: Relate to the given equation
We know from the problem that:
\[ P(X \leq 2) = \frac{k}{2^{16}} \]
Thus,
\[ \frac{137}{65536} = \frac{k}{65536} \]
This implies \( k = 137 \).

### Step 10: Calculate \( \frac{k - 47}{10} \)
Now, we need to find:
\[ \frac{k - 47}{10} = \frac{137 - 47}{10} = \frac{90}{10} = 9 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{9} \]