What is the probability that 4 will appear exactly thrice on rolling a normal dice four times ?

Options

A. `(1)/(324)`
B. `(5)/(1296)`
C. `(1)/(216)`
D. `(5)/(324)`

Answer

To solve the problem of finding the probability that the number 4 will appear exactly three times when rolling a normal die four times, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Probability of Rolling a 4:**
The probability of rolling a 4 on a single die is:
\[ P(4) = \frac{1}{6} \]

2. **Identify the Probability of Not Rolling a 4:**
The probability of rolling any number other than 4 (i.e., 1, 2, 3, 5, or 6) is:
\[ P(\text{not } 4) = 1 - P(4) = 1 - \frac{1}{6} = \frac{5}{6} \]

3. **Determine the Required Outcomes:**
We want to find the probability of getting exactly three 4s in four rolls. This means we need three rolls to be 4 and one roll to be a number other than 4.

4. **Calculate the Probability of a Specific Arrangement:**
The probability of a specific arrangement (e.g., 4, 4, 4, not 4) is given by:
\[ P(4, 4, 4, \text{not } 4) = P(4) \times P(4) \times P(4) \times P(\text{not } 4 = \frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} \times \frac{5}{6} \]
This simplifies to:
\[ P(4, 4, 4, \text{not } 4) = \left(\frac{1}{6}\right)^3 \times \left(\frac{5}{6}\right) = \frac{1}{216} \times \frac{5}{6} = \frac{5}{1296} \]

5. **Count the Number of Favorable Arrangements:**
The number of ways to arrange three 4s and one not-4 in four rolls can be calculated using combinations:
\[ \text{Number of arrangements} = \binom{4}{3} = 4 \]
This is because we are choosing 3 positions out of 4 for the number 4.

6. **Calculate the Total Probability:**
Now, we multiply the probability of one specific arrangement by the number of arrangements:
\[ P(\text{exactly 3 fours}) = \text{Number of arrangements} \times P(4, 4, 4, \text{not } 4) = 4 \times \frac{5}{1296} = \frac{20}{1296} \]
Simplifying this gives:
\[ P(\text{exactly 3 fours}) = \frac{5}{324} \]

### Final Answer:
The probability that 4 will appear exactly three times when rolling a normal die four times is:
\[ \frac{5}{324} \]