If you have a four sided die, and every time you roll a 4 you get roll again and add it to your previous total, what are the chances you get a total of 12.

Let's break down the problem to calculate the probability of getting a total of 12 when rolling a four-sided die (d4) with the rule that every time you roll a 4, you roll again and add it to your previous total.

### Key Points:
1. **Possible Rolls**: On a d4, you can roll a 1, 2, 3, or 4.
2. **Rule**: If you roll a 4, you get to roll again, and you keep adding until you either roll something other than 4 or reach your target total.

### Pathways to Total 12:
To reach exactly 12, you must consider the series of rolls that add up to this total. Here are the possible sequences:

1. **Three 4s** followed by a roll of 1:  
   \( 4 + 4 + 4 + 1 = 12 \)

2. **Two 4s** followed by a roll of 3:  
   \( 4 + 4 + 3 + 1 = 12 \)

### Probability Calculation:
- **Probability of rolling a 4**: \(\frac{1}{4}\)
- **Probability of rolling a 1, 2, or 3**: \(\frac{1}{4}\) each

Let's calculate the probability of each pathway:

1. **Pathway 1** (Three 4s and a 1):
   - Probability = \(\left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) = \frac{1}{256}\)

2. **Pathway 2** (Two 4s, a 3, and a 1):
   - Probability = \(\left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) = \frac{1}{256}\)

Finally, since these events are mutually exclusive, we add their probabilities together:

\[
\text{Total Probability} = \frac{1}{256} + \frac{1}{256} = \frac{2}{256} = \frac{1}{128}
\]

So, the probability of getting a total of 12 is \(\boxed{\frac{1}{128}}\).

Would you like more details or have any other questions?

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Here are some related questions to explore further:

1. What is the probability of getting a total of 8 with the same rules?
2. How does the probability change if the die has 6 sides instead of 4?
3. What is the expected value of a roll in this scenario?
4. How would the probabilities change if you could roll indefinitely until you reach or exceed a total of 12?
5. Can you generalize the probability for any total greater than 12?

**Tip:** When calculating probabilities in dice games with rerolls, consider all possible sequences that lead to the desired outcome. This often involves adding up probabilities of different valid pathways.

Calculate the probability of rolling a total of 12 with a four-sided die where rolling a 4 allows a reroll. Explore the pathways to reaching this total and understand the probabilities involved. Detailed solution and step-by-step calculation provided.

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