To determine the probability $P(\text{five})$ of rolling a total of five with two dice, let’s analyze the sample space in the image. The image shows all 36 possible outcomes of rolling two six-sided dice.

Step-by-Step Solution

1. Count All Possible Outcomes: There are 36 total outcomes (since each die has 6 faces and $6 \times 6 = 36$).

2. Identify Outcomes That Total Five: For a total of five, the possible outcomes (pairs of dice rolls) are:

   • (1,4)
   • (2,3)
   • (3,2)
   • (4,1)

   Each of these pairs represents a way to achieve a sum of five, giving us 4 favorable outcomes.

3. Calculate Probability: The probability $P(\text{five})$ is the number of favorable outcomes divided by the total number of outcomes: $P(\text{five}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{4}{36} = \frac{1}{9}$

Final Answer:

The probability of rolling a total of five is $\frac{1}{9}$.

Would you like a deeper explanation on how probabilities are calculated for dice or have any questions?

Related Questions to Explore:

1. What is the probability of rolling a sum of seven with two dice?
2. How many outcomes give a sum of eight with two dice?
3. What is the probability of rolling doubles (e.g., (1,1), (2,2), etc.)?
4. How can we calculate probabilities for sums greater than 7 with two dice?
5. What are the odds of rolling an even sum?

Tip: For problems involving dice, listing all possible outcomes systematically can help identify favorable outcomes accurately.