Two symmetric dice flipped with each dice having two side painted red, two painted black, one painted yellow and the other painted white. What is the probability that both land on the same ?

Options

A. `(3)/(18)`
B. `(2)/(9)`
C. `(5)/(18)`
D. `(1)/(3)`

Explanation

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Dice Configuration
Each die has:
- 2 sides painted Red (R)
- 2 sides painted Black (B)
- 1 side painted Yellow (Y)
- 1 side painted White (W)

So, the possible outcomes for each die are: R, R, B, B, Y, W.

### Step 2: Calculate Total Outcomes
When flipping two dice, the total number of outcomes is given by the product of the outcomes for each die. Since each die has 6 sides, the total number of outcomes is:
\[ \text{Total Outcomes} = 6 \times 6 = 36 \]

### Step 3: Identify Favorable Outcomes
Next, we need to find the number of favorable outcomes where both dice show the same color. The possible combinations for the same color are:
1. Both dice show Red (R, R)
2. Both dice show Black (B, B)
3. Both dice show Yellow (Y, Y)
4. Both dice show White (W, W)

Now, let's count the favorable outcomes:
- For Red: There are 2 ways to choose Red from the first die and 2 ways from the second die, giving us \(2 \times 2 = 4\) combinations.
- For Black: Similarly, there are \(2 \times 2 = 4\) combinations.
- For Yellow: There is only 1 way to choose Yellow from each die, giving us \(1 \times 1 = 1\) combination.
- For White: There is only 1 way to choose White from each die, giving us \(1 \times 1 = 1\) combination.

Now, we sum these favorable outcomes:
\[ \text{Favorable Outcomes} = 4 \, (\text{Red}) + 4 \, (\text{Black}) + 1 \, (\text{Yellow}) + 1 \, (\text{White}) = 10 \]

### Step 4: Calculate the Probability
The probability of both dice landing on the same color is given by the ratio of favorable outcomes to total outcomes:
\[ \text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{10}{36} \]

### Step 5: Simplify the Probability
We can simplify \(\frac{10}{36}\):
\[ \frac{10}{36} = \frac{5}{18} \]

### Conclusion
Thus, the required probability that both dice land on the same color is:
\[ \frac{5}{18} \]