We toss a fair six-sided die twice and the possible outcomes are (i,j) where i and j are numbers 1, 2, 3, 4, 5, 6 corresponding to the side that turns up. What is the probability that i + j = 8 ? Round your answer with exactly four decimal places.



We are tasked with finding the probability that the sum of the two dice rolls, \(i + j\), equals 8. The possible outcomes from rolling two fair six-sided dice are pairs \((i, j)\), where \(i\) and \(j\) are numbers from 1 to 6. Let's break it down step by step:

### Total Possible Outcomes
Since each die has 6 sides, there are a total of \(6 \times 6 = 36\) possible outcomes when rolling two dice.

### Favorable Outcomes
We need to count how many pairs \((i, j)\) satisfy the equation \(i + j = 8\). The possible pairs that sum to 8 are:

\[
(2,6), (3,5), (4,4), (5,3), (6,2)
\]

These are the five pairs where the sum is exactly 8.

### Probability Calculation
The probability of an event is the ratio of favorable outcomes to the total number of possible outcomes. Thus, the probability is:

\[
P(i + j = 8) = \frac{5}{36}
\]

Now, dividing and rounding to four decimal places:

\[
P(i + j = 8) = \frac{5}{36} \approx 0.1389
\]

### Final Answer
The probability that \(i + j = 8\) is \( \boxed{0.1389} \).

Would you like further details or have any other questions?

Here are 5 related questions for deeper understanding:
1. What is the probability that \(i + j = 7\) in the same scenario?
2. How many outcomes result in a sum less than 6?
3. What is the probability that both dice show the same number?
4. If the dice were not fair, how would the probability calculation change?
5. How does the number of favorable outcomes change for different sums?

**Tip:** When working with probability problems, always start by calculating the total number of possible outcomes.

We toss a fair six-sided die twice and the possible outcomes are (i,j) where i and j are numbers 1, 2, 3, 4, 5, 6 corresponding to the side that turns up. What is the probability that i + j = 8 ? Round your answer with exactly four decimal places.

Discover how to calculate the probability of getting a sum of 8 when tossing two fair six-sided dice. With a total of 36 possible outcomes, find the 5 favorable pairs and use basic probability to solve this problem. The solution is suitable for students in Grades 6-8.

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