To solve this problem, let's break it down into a series of steps.

Step 1: Understand the Problem

We need to find the probability of rolling doubles on a pair of dice, given that the sum of the two dice is less than 11.

Step 2: Identify Possible Outcomes

1. Total Possible Outcomes for Rolling Two Dice: There are $6 \times 6 = 36$ total outcomes for any roll of two dice since each die has 6 faces.
2. Condition - Sum Less Than 11: First, let’s find the outcomes where the sum of the dice is less than 11. These sums range from 2 to 10.

Outcomes for Sum Less Than 11

The sums that are possible under this condition are as follows:

• Sum of 2: (1,1)
• Sum of 3: (1,2), (2,1)
• Sum of 4: (1,3), (2,2), (3,1)
• Sum of 5: (1,4), (2,3), (3,2), (4,1)
• Sum of 6: (1,5), (2,4), (3,3), (4,2), (5,1)
• Sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
• Sum of 8: (2,6), (3,5), (4,4), (5,3), (6,2)
• Sum of 9: (3,6), (4,5), (5,4), (6,3)
• Sum of 10: (4,6), (5,5), (6,4)

Counting these, we have a total of 27 outcomes where the sum is less than 11.

Step 3: Identify Favorable Outcomes (Doubles)

Doubles occur when both dice show the same number. Within the outcomes where the sum is less than 11, the doubles are:

• Sum of 2: (1,1)
• Sum of 4: (2,2)
• Sum of 6: (3,3)
• Sum of 8: (4,4)
• Sum of 10: (5,5)

So, there are 5 outcomes where doubles occur under the condition that the sum is less than 11.

Step 4: Calculate the Probability

The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes where the sum is less than 11.

$P(\text{doubles} \mid \text{sum} < 11) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes where sum < 11}} = \frac{5}{27}$

Step 5: Round the Answer

$\frac{5}{27} \approx 0.185$

Thus, the probability that doubles are rolled, given that the sum is less than 11, is approximately 0.185 (rounded to three decimal places).

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Would you like more details on any of these steps? Here are some related questions to explore further:

1. How would the probability change if the sum was less than 10 instead of 11?
2. What is the probability of rolling a sum greater than or equal to 7 on two dice?
3. How many total outcomes have a sum of exactly 9 on a pair of dice?
4. What is the probability of rolling doubles on two dice without any condition?
5. How does changing to a pair of three-sided dice affect the probability calculations?

Tip: Knowing how to set up a probability space (like the total possible outcomes in dice rolls) is essential for solving conditional probability questions!