Problem 42 - Algebra and Trigonometry

Question

Two fair dice are rolled. Determine the probability that the sum of the faces is \(11 .\)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{18} \).

Step 1: Determine Total Possible Outcomes

When two fair dice are rolled, each die has 6 faces. Therefore, the total number of possible outcomes is calculated as follows: \(6 \times 6 = 36\). This means there are 36 possible outcomes when rolling two dice.

Step 2: Identify Favorable Outcomes

To find the probability that the sum of the faces is 11, identify all possible pairs of die rolls that add up to 11. These pairs are (5, 6) and (6, 5). So, there are 2 favorable outcomes.

Step 3: Calculate the Probability

The probability is the number of favorable outcomes divided by the total number of possible outcomes. Therefore, the probability is: \[ P = \frac{2}{36} = \frac{1}{18} \].

Key Concepts

Dice Rolls

Rolling dice is a common way to introduce the concept of probability. A die is a small, typically cube-shaped object with each of its six faces showing a different number of dots (from 1 to 6). When we roll a die, it lands randomly on one of these faces.

When rolling two dice, the outcome of each die roll is independent of the other. This means the result of one die does not affect the result of the other. There are 6 possibilities for the first die and 6 for the second, making it 36 possible outcomes in total. This independence and the fixed number of faces make calculating probabilities easier as we can use a combinatorial approach.

Favorable Outcomes

Favorable outcomes are the specific outcomes that satisfy the condition of the probability question. In this exercise, we want the sum of the faces to be 11.

We can list out the combinations of numbers on two dice that add up to 11. These pairs are:

• (5, 6)
• (6, 5)

These pairs are the favorable outcomes because they are the only ones among all the possible outcomes that make the sum 11. Always list out and count these carefully to ensure you have all the favorable pairs.

Total Possible Outcomes

To understand the total possible outcomes, we need to think about all the ways you can roll two dice.

Each die has 6 faces. Therefore, the first die can land on any of 6 faces, and the same goes for the second die. We can use multiplication to find the total outcomes: \[6 \times 6 = 36\]\
This means there are 36 possible combinations of numbers that can appear when two dice are rolled. This fixed total number of outcomes is important, as it forms the denominator when calculating probability.

Calculating Probability

To calculate probability, we use the ratio of favorable outcomes to total possible outcomes. The formula is: \[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Possible Outcomes}} \]

From the previous steps, we determined there are 2 favorable outcomes and 36 total possible outcomes. Plugging these numbers into our formula, we get:
\[\text{P} = \frac{2}{36} = \frac{1}{18} \]

The probability that the sum of the faces is 11 is thus: \[\frac{1}{18}\].
Remember, probability is always a value between 0 and 1, where 0 means the event cannot happen and 1 means it is certain to happen.