To solve the problem of finding the probability of obtaining a sum greater than 3 but not exceeding 6 when rolling two fair dice, we can follow these steps:
### Step 1: Determine the Sample Space
When rolling two dice, each die has 6 faces. Therefore, the total number of possible outcomes when rolling two dice is:
\[
\text{Total outcomes} = 6 \times 6 = 36
\]
### Step 2: Identify the Favorable Outcomes
Next, we need to find the outcomes where the sum of the two dice is greater than 3 but not exceeding 6. We can list the possible sums:
- **Sum = 4**: The combinations are (1,3), (2,2), (3,1).
- **Sum = 5**: The combinations are (1,4), (2,3), (3,2), (4,1).
- **Sum = 6**: The combinations are (1,5), (2,4), (3,3), (4,2), (5,1).
Now, let's count the total number of favorable outcomes:
- For sum = 4: 3 outcomes
- For sum = 5: 4 outcomes
- For sum = 6: 5 outcomes
Adding these together gives us:
\[
\text{Total favorable outcomes} = 3 + 4 + 5 = 12
\]
### Step 3: Calculate the Probability
The probability \( P \) of an event is given by the ratio of the number of favorable outcomes to the total number of outcomes. Therefore, we can calculate the probability as follows:
\[
P(\text{sum > 3 and } \leq 6) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{12}{36} = \frac{1}{3}
\]
### Final Answer
Thus, the probability of obtaining a sum greater than 3 but not exceeding 6 when rolling two fair dice is:
\[
\frac{1}{3}
\]
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