Problem 18 - Algebra for College Students

Question

Find the probability of rolling each of the following events:A sum of \(11 \quad \frac{1}{18}\)

Step-by-Step Solution

Answer

The probability of rolling a sum of 11 is \( \frac{1}{18} \).

Step 1: Determine Total Possible Outcomes

When rolling two six-sided dice, each die has 6 faces. Therefore, the total number of outcomes when rolling two dice is calculated by multiplying the number of outcomes of the first die by those of the second die: \[ 6 \times 6 = 36 \] So, there are 36 possible outcomes when rolling two dice.

Step 2: Identify Successful Outcomes for a Sum of 11

To find the successful outcomes that result in a sum of 11, list the pairs of numbers on two dice that add up to 11. These pairs are (5, 6) and (6, 5). Therefore, there are 2 successful outcomes.

Step 3: Calculate the Probability

The probability of an event is given by the formula:\[ \text{Probability} = \frac{\text{Number of Successful Outcomes}}{\text{Total Possible Outcomes}} \]Substitute the values:\[ \text{Probability of a sum of 11} = \frac{2}{36} = \frac{1}{18} \] This is the probability of rolling a sum of 11 with two dice.

Key Concepts

Rolling Dice

Rolling dice is a common practice in probability theory because it provides a straightforward example of calculating probabilities. When you roll a standard six-sided die, it can land on any one of the six numbers: 1, 2, 3, 4, 5, or 6. This is important because each side has an equal chance of appearing, which makes the situation a uniform probability model.

When rolling two dice, each die operates independently, meaning the result of one does not affect the result of the other. This independence allows us to determine the total number of possible outcomes by multiplying the number of outcomes for each die. Since each die has 6 possible outcomes, when rolling two dice together, we have:

• First die: 6 outcomes
• Second die: 6 outcomes
• Total outcomes: 6 * 6 = 36

Understanding this basic setup is crucial for any probability calculations involving dice.

Event Outcomes

Event outcomes refer to the specific results we are interested in when an experiment, like rolling dice, is conducted. In our exercise, we are looking for a particular event: rolling two dice and getting a sum of 11.

To determine which specific rolls result in a sum of 11, we list the possible pairs:

• (5, 6)
• (6, 5)

There are two possible successful outcomes from the 36 total outcomes. Recognizing these specific pairs is essential because it defines what success looks like for the event we are calculating the probability for. Understanding event outcomes helps us prioritize which results we should focus on for further calculations.

Probability Calculation

Probability calculation involves determining how likely an event is to occur. In our example of rolling two dice, we're interested in finding the probability of the event where the sum is equal to 11.

The formula for probability is:

\[ \text{Probability} = \frac{\text{Number of Successful Outcomes}}{\text{Total Possible Outcomes}} \]

By substituting the relevant numbers from our example:

• Number of Successful Outcomes: 2 (from pairs (5, 6) and (6, 5))
• Total Possible Outcomes: 36 (from rolling two six-sided dice)
• Probability: \[ \frac{2}{36} = \frac{1}{18} \]

This means the probability of rolling a sum of 11 when rolling two dice is \( \frac{1}{18} \). This simple yet effective formula allows us to calculate the chance of almost any event, provided we have the necessary information about possible outcomes.