Question

The following questions are from ARTIST (reproduced with permission) You roll 2 fair six-sided dice. Which of the following outcomes is most likely to occur on the next roll? A. Getting double \(3 .\) B. Getting a 3 and a 4. C. They are equally likely. Explain your choice.

Short answer

Option B (getting a 3 and a 4) is more likely.

Step-by-step solution

Understanding Possible Outcomes

When rolling two six-sided dice, each die has 6 faces, so there are a total of 36 possible outcomes (6 sides on the first die times 6 sides on the second die).

Calculating the Probability for Double 3

To get a double 3, both dice must land on 3, which is just one specific outcome: (3,3). The probability for this is \( \frac{1}{36} \) because there is only one way to roll double 3 out of 36 possible outcomes.

Calculating the Probability for Getting a 3 and a 4

To get a 3 and a 4, there are two possible outcomes: (3,4) or (4,3). The probability for each is \( \frac{1}{36} \), so the total probability for getting a 3 and a 4 in any order is \( \frac{2}{36} = \frac{1}{18} \).

Comparing the Probabilities

The probability of getting double 3 is \( \frac{1}{36} \), and the probability of getting a 3 and a 4 is \( \frac{1}{18} \), which is greater than \( \frac{1}{36} \).

Conclusion

Since the probability of getting a 3 and a 4 is higher than that of getting double 3, option B is more likely than option A. They are not equally likely.

Key concepts

Understanding Dice Probabilities

When you roll two six-sided dice, each die provides 6 possible outcomes, leading to a combination of 36 outcomes in total. This scenario showcases the beauty of probability theory where each distinct outcome, such as rolling a specific number on both dice, is equally likely. To understand how likely a specific outcome is, we consider the number of favorable outcomes over the total possible outcomes.

• Rolling any pair of numbers, like a double 3 (which is (3,3)), is just one out of these 36 outcomes, making the probability of a double 3 exactly \( \frac{1}{36} \).
• For combinations like rolling a 3 and a 4, you consider both possible sequences: (3,4) and (4,3). Together they occur in \( \frac{2}{36} \) or \( \frac{1}{18} \) probability.

Understanding these simple probability fractions can help predict the outcome of dice rolls accurately. Recognizing the simplicity of this process is crucial in grasping probability theory.

Combinatorial Analysis in Dice Rolling

Combinatorial analysis is a mathematical technique used to count or combine different outcomes. In the context of dice, it helps analyze the possible results of throwing dice. With two six-sided dice, combinatorial analysis allows us to explore and count the outcomes that interest us.

• For a single result like (3,3), there's only one combination, so we count it as just one way.
• When analyzing scenarios like rolling a 3 and a 4, combinatorial analysis reveals multiple orders of achieving that pair: (3,4) and (4,3). This results in two combinations, providing a higher probability than a single outcome like a double 3.

By using combinatorial analysis, one can systematically verify and infer probabilities, highlighting the practical use of this analysis in games and probability questions involving dice.

Principle of Equal Likelihood

Equal likelihood, a fundamental principle in probability theory, asserts that each outcome of a probabilistic event has the same chance of occurring, assuming there's no bias. In scenarios like those of rolling fair dice, this principle greatly simplifies analysis and predictions.

• Each side or face of a die is equally likely to land facing up, with a 1 in 6 chance for a single die roll.
• For two dice, every unique pair of numbers from 1 through 6 happens with a 1 in 36 chance.

This principle helps mathematicians and analysts predict and compare the likelihoods of different events. By ensuring fairness, or equal probability, for each dice outcome, probability calculations become straightforward, offering clear insights into expected results.