Problem 44 - Algebra and Trigonometry

Question

Two fair dice are rolled. Determine the probability that the sum of the faces is \(12 .\)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{36} \).

Step 1 - Identify all possible outcomes

Each die has 6 faces, so the total number of outcomes when rolling two dice is: \[6 \times 6 = 36\]

Step 2 - Determine the desired outcome

To achieve a sum of 12, both dice must show a 6. There is only one combination that satisfies this condition: (6, 6).

Step 3 - Calculate the probability

The probability is the number of favorable outcomes divided by the total number of possible outcomes: \[\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{36}\]

Key Concepts

probability calculations

Probability calculations form the foundation of understanding how likely an event is to occur. In probability theory, we often express probability as a ratio of favorable outcomes to the total number of possible outcomes. For example, when rolling two dice, every possible result is an outcome. An event, such as getting a sum of 12, is a specific combination of these outcomes.
To calculate the probability of such an event, follow these steps:

• Identify the total number of possible outcomes. For two dice, this is calculated by multiplying the number of faces on each die: 6 faces per die, so \(6 \times 6 = 36\) total outcomes.
• Determine the number of favorable outcomes. For a sum of 12, there is only one favorable outcome (rolling two 6s).
• Divide the number of favorable outcomes by the total number of possible outcomes to get the probability. Here, it’s \(\frac{1}{36}\).

This step-by-step breakdown ensures precision in calculating the probability of any desired event.

sum of dice

The sum of dice is a popular concept in probability problems, especially in games involving chance, like board games or gambling. When rolling two dice, each face showing from 1 to 6 results in sums ranging from 2 (\(1 + 1\)) to 12 (\(6 + 6\)).
Understanding how sums are made up helps in calculating probabilities. For example, there’s only one way to achieve a sum of 2 (both dice showing 1) or 12 (both dice showing 6). Some sums, like 7, have more combinations (\(1+6, 2+5, 3+4, etc.\)). This is important because it affects how often these sums occur.
Therefore, breaking down sums into their possible dice combinations aids in better understanding probability distributions for various sums.

favorable outcomes

Favorable outcomes are specific results that meet the criteria of an event we are interested in. In our exercise, the event is getting a sum of 12 from rolling two dice.
We define favorable outcomes by first identifying all possible outcomes and then isolating those that meet our specific conditions. Hence, a favorable outcome for the sum of 12 is the pair (6, 6).

• Start by listing all potential outcomes – for two dice, there are 36 possible pairs.
• Identify the pairs that match the desired criterion – in this case, just one pair: (6, 6).
• This single favorable outcome is used to calculate the probability of the event occurring.

Recognizing and counting favorable outcomes correctly is crucial for accurate probability calculations, ensuring clarity in understanding the likelihood of events.