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

Options

A. `(3)/(16)`
B. `(2)/(9)`
C. `(5)/(18)`
D. `(1)/(3)`

Answer

To solve the problem of finding the probability that both symmetric dice land on the same color, we will follow these steps:

### Step 1: Identify the colors on the dice
Each die has:
- 2 sides painted red
- 2 sides painted black
- 1 side painted yellow
- 1 side painted white

Thus, the total colors available on each die are red, black, yellow, and white.

### Step 2: Calculate the total outcomes
When two dice are rolled, each die has 6 faces. Therefore, the total number of outcomes when rolling two dice is:
\[ \text{Total Outcomes} = 6 \times 6 = 36 \]

### Step 3: Calculate the probability of both dice landing on the same color
We will calculate the probability for each color:

1. **Red**:
- Probability of the first die landing on red = \( \frac{2}{6} = \frac{1}{3} \)
- Probability of the second die landing on red = \( \frac{2}{6} = \frac{1}{3} \)
- Combined probability for both dice landing on red:
\[ P(\text{Both Red}) = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} \]

2. **Black**:
- Probability of the first die landing on black = \( \frac{2}{6} = \frac{1}{3} \)
- Probability of the second die landing on black = \( \frac{2}{6} = \frac{1}{3} \)
- Combined probability for both dice landing on black:
\[ P(\text{Both Black}) = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} \]

3. **Yellow**:
- Probability of the first die landing on yellow = \( \frac{1}{6} \)
- Probability of the second die landing on yellow = \( \frac{1}{6} \)
- Combined probability for both dice landing on yellow:
\[ P(\text{Both Yellow}) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \]

4. **White**:
- Probability of the first die landing on white = \( \frac{1}{6} \)
- Probability of the second die landing on white = \( \frac{1}{6} \)
- Combined probability for both dice landing on white:
\[ P(\text{Both White}) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \]

### Step 4: Sum the probabilities of landing on the same color
Now, we sum the probabilities of both dice landing on red, black, yellow, and white:
\[ P(\text{Same Color}) = P(\text{Both Red}) + P(\text{Both Black}) + P(\text{Both Yellow}) + P(\text{Both White}) \]
\[ P(\text{Same Color}) = \frac{1}{9} + \frac{1}{9} + \frac{1}{36} + \frac{1}{36} \]

To add these fractions, we need a common denominator. The least common multiple of 9 and 36 is 36:
\[ P(\text{Same Color}) = \frac{4}{36} + \frac{4}{36} + \frac{1}{36} + \frac{1}{36} = \frac{4 + 4 + 1 + 1}{36} = \frac{10}{36} = \frac{5}{18} \]

### Final Answer
The probability that both dice land on the same color is:
\[ \frac{5}{18} \]

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