I have two six-sided dice, each with faces numbered from 1 to 6. One of the dice is fair, but the other is not – it will land on numbers 1 to 5 with equal probability, but lands on 6 with a different probability.
When I roll the dice the probability that I get a total of 12 is . What is the probability that I get a total of 2 when I roll the dice?

To solve this problem, let's break it down into steps.

### Step 1: Define the probabilities
Let:
- \( p \) be the probability that the biased die lands on 6.
- The fair die has equal probabilities of landing on any of its 6 faces, so each face has a probability of \( \frac{1}{6} \).

### Step 2: Calculate the probability of getting a total of 12
A total of 12 can only be obtained if both dice show 6. Therefore, the probability of rolling a total of 12 is:
\[
P(\text{Total} = 12) = \left(\frac{1}{6}\right) \times p
\]

This probability is given as \(\frac{1}{18}\), so:
\[
\frac{1}{6} \times p = \frac{1}{18}
\]
Solving for \( p \):
\[
p = \frac{1}{18} \times 6 = \frac{1}{3}
\]
Thus, the probability that the biased die lands on 6 is \( \frac{1}{3} \).

### Step 3: Calculate the probability of getting a total of 2
A total of 2 can only occur if both dice show 1. Since the fair die has a probability of \( \frac{1}{6} \) of landing on 1 and the biased die has a probability of \( \frac{1}{6} \) of landing on 1 (because it lands on 1 to 5 with equal probability), the probability of rolling a total of 2 is:
\[
P(\text{Total} = 2) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}
\]

### Final Answer
The probability that you get a total of 2 when you roll the dice is \( \frac{1}{36} \).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. What is the probability of rolling a total of 7 with these dice?
2. If the biased die lands on 6 with a probability of 0.5, what is the probability of rolling a total of 10?
3. How would the probability of getting a total of 4 change if the biased die had a 0.2 probability of landing on 6?
4. What is the probability of rolling an odd number total with these dice?
5. How does the probability distribution change if both dice are biased in the same way?

**Tip:** When dealing with non-standard dice or biased dice, start by defining the probabilities for each outcome clearly before proceeding with calculations.

Explore the probabilities of rolling two six-sided dice, one fair and one biased. Calculate the probability of getting a total of 2 and 12, with detailed step-by-step solutions. Learn about probability theory applied to dice rolling. Suitable for Grades 10-12.

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