Question

If a pair of dice is rolled, what is the probability of rolling a 2 or a 3 on at least one die? A. \(\frac{1}{6}\) B. \(\frac{4}{9}\) C. \(\frac{5}{9}\) D. \(\frac{2}{3}\)

Short answer

C. \( \frac{5}{9} \)

Step-by-step solution

- Determine total possible outcomes

A pair of dice has 6 faces each, leading to a total of 6 * 6 = 36 possible outcomes.

- Identify favorable outcomes

Find the combinations where at least one die shows a 2 or a 3. These include: (2,x), (3,x), (x,2), and (x,3) where x can be any number from 1 to 6.

- List outcomes and avoid double counting

List the outcomes: (2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(1,2),(4,2),(5,2),(6,2),(1,3),(4,3),(5,3),(6,3). There are 20 such outcomes.

- Calculate the probability

The probability is the ratio of favorable outcomes to the total possible outcomes: \( P(\text{at least one die shows 2 or 3}) = \frac{20}{36} = \frac{5}{9} \).

Key concepts

Probability of Rolling Specific Outcomes

Probability is a measure of how likely an event is to occur. In dice games, you often calculate probability to understand your chances of rolling specific outcomes. With a pair of dice, each die has 6 faces showing numbers from 1 to 6. The total number of possible outcomes when rolling two dice is found by multiplying the number of faces on one die by the number of faces on the other die: \(6 \times 6 = 36\). Each of these outcomes is equally likely.

Counting Favorable Dice Combinations

When calculating the probability of rolling a specific outcome, such as a 2 or a 3 on at least one die, you need to list all favorable combinations. In our scenario, any combination where at least one die shows a 2 or a 3 is favorable. Start by writing combinations where one die is fixed at either 2 or 3, and the other die can be any number from 1 to 6, this includes: \(2,x\) and \(3,x\). You'll get outcomes like \(2,1\), \(2,2\), \(2,3\) and so on up to \(2,6\). Then do the same for 3: \(3,1\), \(3,2\), etc. Similarly, consider patterns where the first die is any number, and the second die is a 2 or a 3: \(x,2\) and \(x,3\).

Probability and GED Math

Understanding how to calculate probabilities is essential for GED math. It's a topic that helps in real-life decision-making and in exams. When performing these calculations, it is fundamental not to double count identical combinations. For instance, (2,3) and (3,2) are different outcomes, and each should be counted once. In our example, once you've correctly listed all the favorable outcomes without duplication, the probability is calculated as the number of favorable outcomes divided by the total possible outcomes. Therefore, \(P(at \text{ least one die shows 2 or 3}) = \frac{20}{36} \) simplifies to \( \frac{5}{9} \). For the GED math, being able to navigate these steps with understanding is crucial for success.