A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is`K/(3^11)`, then k is equal to

Options

A. `82`
B. `75`
C. `164`
D. `123`

Answer

To solve the problem, we need to find the probability of getting at least 4 successes when a pair of dice is thrown 5 times, where a success is defined as rolling a total of 5.

### Step-by-step Solution:

1. **Determine the probability of success (p)**:
- A pair of dice can produce a total of 5 in the following ways:
- (1, 4)
- (2, 3)
- (3, 2)
- (4, 1)
- Thus, there are 4 successful outcomes.
- The total number of outcomes when rolling two dice is \(6 \times 6 = 36\).
- Therefore, the probability of getting a total of 5 in a single throw is:
\[ p = \frac{4}{36} = \frac{1}{9} \]

2. **Determine the probability of failure (q)**:
- The probability of not getting a total of 5 is:
\[ q = 1 - p = 1 - \frac{1}{9} = \frac{8}{9} \]

3. **Calculate the probability of at least 4 successes**:
- We need to find \(P(X \geq 4)\), which can be calculated as:
\[ P(X \geq 4) = P(X = 4) + P(X = 5) \]
- Using the binomial probability formula:
\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]
- For \(P(X = 4)\):
\[ P(X = 4) = \binom{5}{4} p^4 q^1 = \binom{5}{4} \left(\frac{1}{9}\right)^4 \left(\frac{8}{9}\right)^1 \]
\[ = 5 \cdot \left(\frac{1}{9}\right)^4 \cdot \left(\frac{8}{9}\right) = 5 \cdot \frac{1}{6561} \cdot \frac{8}{9} = \frac{40}{59049} \]

- For \(P(X = 5)\):
\[ P(X = 5) = \binom{5}{5} p^5 q^0 = \binom{5}{5} \left(\frac{1}{9}\right)^5 \cdot 1 \]
\[ = 1 \cdot \left(\frac{1}{9}\right)^5 = \frac{1}{59049} \]

4. **Combine the probabilities**:
- Now, we can add the two probabilities:
\[ P(X \geq 4) = P(X = 4) + P(X = 5) = \frac{40}{59049} + \frac{1}{59049} = \frac{41}{59049} \]

5. **Express the probability in the required format**:
- We need to express this probability in the form \( \frac{K}{3^{11}} \).
- Notice that \(59049 = 3^{10}\), so:
\[ P(X \geq 4) = \frac{41}{3^{10}} \]
- To express it in terms of \(3^{11}\):
\[ P(X \geq 4) = \frac{41 \cdot 3}{3^{11}} = \frac{123}{3^{11}} \]

6. **Identify the value of K**:
- From the expression \( \frac{K}{3^{11}} \), we can see that \( K = 123 \).

### Final Answer:
Thus, the value of \( k \) is \( \boxed{123} \).