Four dice are thrown simultaneously . Find the probability that :
All of them show the same face .

Options

A. `(1)/( 216)`
B. `(15)/(16)`
C. `(15)/(36)`
D. `1/2`

Answer

To find the probability that all four dice show the same face when thrown simultaneously, we can follow these steps:

### Step 1: Determine the Total Outcomes
When four dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when throwing four dice is calculated as follows:

\[ \text{Total Outcomes} = 6^4 \]

Calculating this gives:

\[ 6^4 = 1296 \]

### Step 2: Determine the Favorable Outcomes
The favorable outcomes are the scenarios where all four dice show the same face. There are 6 faces on a die (1 through 6), so the favorable outcomes are:

\[ \text{Favorable Outcomes} = 6 \]

This is because all four dice can show either all 1s, all 2s, all 3s, all 4s, all 5s, or all 6s.

### Step 3: Calculate the Probability
The probability \( P \) of an event is given by the formula:

\[ P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} \]

Substituting the values we found:

\[ P(E) = \frac{6}{1296} \]

### Step 4: Simplify the Probability
To simplify \( \frac{6}{1296} \):

\[ P(E) = \frac{1}{216} \]

### Final Answer
Thus, the probability that all four dice show the same face is:

\[ \boxed{\frac{1}{216}} \]

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