Problem 18 - Algebra and Trigonometry

Question

A pair of dice is rolled, and the numbers showing are observed. (a) List the sample space of this experiment. (b) Find the probability of getting a sum of \(7 .\) (c) Find the probability of getting a sum of \(9 .\) (d) Find the probability that the two dice show doubles (the same number). (e) Find the probability that the two dice show different numbers. (f) Find the probability of getting a sum of 9 or higher.

Step-by-Step Solution

Answer

(a) 36 outcomes (b) \(\frac{1}{6}\) (c) \(\frac{1}{9}\) (d) \(\frac{1}{6}\) (e) \(\frac{5}{6}\) (f) \(\frac{5}{18}\)

Step 1: Understanding the Sample Space

A pair of dice can show any number from 1 to 6 on each die. Therefore, the sample space consists of all the possible outcomes of rolling these two dice. Since each die has 6 faces, the total number of outcomes is \(6 \times 6 = 36\). List the outcomes as ordered pairs: (1,1), (1,2), ..., (6,6).

Step 2: Probability of Getting a Sum of 7

To find the probability of obtaining a sum of 7, identify all combinations where the sum of the numbers on the two dice equals 7: these are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). There are 6 outcomes that result in a sum of 7. Hence, the probability is \( \frac{6}{36} = \frac{1}{6} \).

Step 3: Probability of Getting a Sum of 9

Find all combinations where the sum of the numbers equals 9: these are (3,6), (4,5), (5,4), and (6,3). There are 4 outcomes, so the probability is \( \frac{4}{36} = \frac{1}{9} \).

Step 4: Probability of Rolling Doubles

Doubles occur when both dice show the same number, i.e., (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). There are 6 such outcomes. The probability is \( \frac{6}{36} = \frac{1}{6} \).

Step 5: Probability of Different Numbers

Subtract the number of outcomes for doubles from the total outcomes to get the outcomes for different numbers: \(36 - 6 = 30\). Thus, the probability is \( \frac{30}{36} = \frac{5}{6} \).

Step 6: Probability of Getting a Sum of 9 or Higher

Identify all outcomes with sums 9, 10, 11, or 12. These are: (3,6), (4,5), (5,4), (6,3), (4,6), (5,5), (6,4), (5,6), (6,5), (6,6) totaling 10 outcomes. Thus, the probability is \( \frac{10}{36} = \frac{5}{18} \).

Key Concepts

Sample Space

When you roll a pair of dice, each die has numbers ranging from 1 to 6. This creates a variety of possible outcomes. The "sample space" of an event includes all possible combinations of these outcomes. Since each die can result in 6 different numbers, the total number of outcomes is

• 6 choices from the first die
• 6 choices from the second die

These choices form 36 combinations overall, which can be represented as ordered pairs: (1,1), (1,2), ..., up to (6,6). Understanding sample space is crucial because it forms the foundation for calculating probabilities by providing a complete list of every possible result of an experiment.
Each outcome has an equal probability of occurring. This means, in the context of rolling dice, each outcome has a \[\frac{1}{36}\] chance of being rolled.

Combinations

While exploring dice outcomes, it's important to understand that combinations come into play when looking for specific conditions - like when two numbers add up to a particular sum. This refers to specific groupings, such as ways to achieve a sum of 7 with dice rolls. The combinations that sum to 7 include

• (1,6)
• (2,5)
• (3,4)
• (4,3)
• (5,2)
• (6,1)

Each is unique in its arrangement of dice rolls, representing different paths to the same total. Recognizing combinations like these helps in determining probabilities. For instance, since there are 6 combinations that create a total of 7, and 36 outcomes in total, the probability of rolling a 7 is \[\frac{6}{36} = \frac{1}{6}\] Understanding combinations not only aids in calculating probabilities for sums but also for other specific conditions like rolling doubles.

Sum of Dice

The sum of numbers on a pair of dice is another vital concept in probability, often relevant in games and problem-solving. To evaluate the probability of a particular sum, you list every possible outcome that achieves it. For a sum of 9, relevant outcomes are

• (3,6)
• (4,5)
• (5,4)
• (6,3)

These 4 combinations provide a sum of 9. Based on the complete 36-outcome roll, the probability calculation becomes \[\frac{4}{36} = \frac{1}{9}\] Similarly, you can calculate probabilities for other sums, such as 9 or higher, by identifying corresponding sums and their combinations. The higher the sum, typically, the fewer the combinations. Understanding the sum of dice is crucial for evaluating odds and doing probability calculations in games and real-life parallels.

Rolling Doubles

Rolling doubles occurs when both dice show the same number, like (1,1), (3,3), or (6,6). In probability terms, rolling doubles is a very specific combination. The outcomes in rolling a pair of dice that result in doubles remain consistent and predictable due to the limited nature of repeated numbers

• (1,1)
• (2,2)
• (3,3)
• (4,4)
• (5,5)
• (6,6)

These 6 outcomes are limited compared to the 36 possibilities in a pair of dice rolls. Thus, the probability of rolling doubles is \[\frac{6}{36} = \frac{1}{6}\] Double rolls are essential for understanding probability models because they balance simplicity (predictable outcomes) with selective rarity, important when determining odds in controlled settings like board games.