A six faced dice is thrown a dozen times. Probability that six is not obtained even one is

Options

A. `((5)/(6))^(12)`
B. `((5)/(6))^(11)`
C. `((1)/(6))^(12)`
D. None of these

Answer

To solve the problem of finding the probability that a six is not obtained even once when a six-faced die is thrown a dozen times, we can follow these steps:

### Step 1: Understand the Problem
We need to find the probability of not rolling a six in 12 throws of a die.

### Step 2: Define the Probability of Rolling a Six
The probability of rolling a six on a fair six-faced die is:
\[ P(\text{rolling a six}) = \frac{1}{6} \]

### Step 3: Define the Probability of Not Rolling a Six
The probability of not rolling a six (which is rolling a one, two, three, four, or five) is:
\[ P(\text{not rolling a six}) = 1 - P(\text{rolling a six}) = 1 - \frac{1}{6} = \frac{5}{6} \]

### Step 4: Calculate the Probability of Not Rolling a Six in 12 Throws
Since the die is thrown 12 times, and we want the probability that a six is not rolled even once, we can use the multiplication rule of independent events. The probability of not rolling a six in each of the 12 throws is:
\[ P(\text{not rolling a six in 12 throws}) = \left( \frac{5}{6} \right)^{12} \]

### Step 5: Final Calculation
Now we can compute the final probability:
\[ P(\text{not rolling a six in 12 throws}) = \left( \frac{5}{6} \right)^{12} \]

### Conclusion
Thus, the probability that a six is not obtained even once in 12 throws of a die is:
\[ \left( \frac{5}{6} \right)^{12} \]