11) In a particular game, the player whose turn it is rolls two standard fair dice, with faces numbered 1 to 6, and can choose the result from either die or the sum of the two results as the score for that turn.
It is Luigi’s turn to play and he needs to score exactly 5 to win the game. What is the probability that Luigi can win the game at this turn?

To determine the probability that Luigi can win the game, we need to calculate the number of ways he can achieve a score of 5 and divide it by the total number of possible outcomes when rolling two dice.

### Possible Ways to Achieve a Score of 5
Luigi can choose either:
1. The number rolled on the first die.
2. The number rolled on the second die.
3. The sum of the two dice.

We need to consider each of these possibilities.

#### 1. Using the first die's result:
- Luigi wins if he rolls a 5 on the first die. The possible outcomes for the second die do not matter.

#### 2. Using the second die's result:
- Luigi wins if he rolls a 5 on the second die. The possible outcomes for the first die do not matter.

#### 3. Using the sum of the two dice:
- The sum of the two dice equals 5 in the following pairs:
  - (1, 4)
  - (2, 3)
  - (3, 2)
  - (4, 1)

### Counting the Winning Outcomes
Let's count the favorable outcomes:
1. **First die is 5**: There are 6 possible outcomes for the second die (1 to 6).
2. **Second die is 5**: There are 6 possible outcomes for the first die (1 to 6).
3. **Sum is 5**: There are 4 specific pairs as listed above.

However, some outcomes may have been counted twice:
- **(5, 5)** was counted both as "first die is 5" and "second die is 5". It should be counted only once.

So, the total number of favorable outcomes is:
\[ 6 + 6 + 4 - 1 = 15 \]

### Total Possible Outcomes
Since Luigi rolls two dice, the total number of possible outcomes is:
\[ 6 \times 6 = 36 \]

### Probability Calculation
The probability that Luigi can win the game is the number of favorable outcomes divided by the total number of outcomes:
\[
P(\text{Winning}) = \frac{15}{36} = \frac{5}{12}
\]

### Final Answer
The probability that Luigi can win the game on this turn is \(\frac{5}{12}\).

Would you like more details or have any questions?  

Here are 5 related questions for further exploration:
1. What is the probability that Luigi rolls a sum of 7?
2. How many ways can Luigi roll a sum greater than 8?
3. What is the probability of rolling doubles with two dice?
4. How does the probability change if Luigi can only choose the sum?
5. If Luigi needs a 6 to win, what is the new probability?

**Tip:** When solving probability problems involving dice, it's helpful to list out all the possible outcomes systematically.

Calculate the probability that Luigi can win the game by scoring exactly 5 on his turn. Explore the possible outcomes of rolling two dice and choosing the result from either die or their sum. Understand the key concepts of probability, dice rolls, and combinatorics with this detailed explanation.

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