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Chapter 4: Problem 19

A pair of dice is to be rolled. In Example \(4.3,\) the probability for each of the possible sums was discussed and three of the probabilities, \(P(2), P(3),\) and \(P(4),\) were found. Find the probability for each of the remaining sums of two dice: \(P(5), P(6), P(7), P(8), P(9), P(10), P(11)\) and \(P(12)\)

Short Answer

Expert verified
The probabilities for each of the remaining sums when rolling two dice are: \(P(5) = \frac{1}{9}\), \(P(6) = \frac{5}{36}\), \(P(7) = \frac{1}{6}\), \(P(8) = \frac{5}{36}\), \(P(9) = \frac{1}{9}\), \(P(10) = \frac{1}{12}\), \(P(11) = \frac{1}{18}\), and \(P(12) = \frac{1}{36}\)

Step by step solution

01

Determining Total Outcomes

When two dice are rolled, there are a total of \(6 * 6 = 36\) possible outcomes as each die has 6 faces.
02

Calculate Probability for Each Sum

To calculate the probability for each sum, it's necessary first to identify the outcomes that result in the desired sum.a. \(P(5)\): The pairs (1,4), (2,3), (3,2), (4,1) result in the sum of 5. So, the probability is \( \frac{4}{36} = \frac{1}{9} \)b. \(P(6)\): The pairs (1,5), (2,4), (3,3), (4,2), (5,1) result in the sum of 6. So, the probability is \( \frac{5}{36} \)c. \(P(7)\): The pairs (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) result in the sum of 7. So, the probability is \( \frac{6}{36} = \frac{1}{6} \)d. \(P(8)\): The pairs (2,6), (3,5), (4,4), (5,3), (6,2) result in the sum of 8. So, the probability is \( \frac{5}{36} \)e. \(P(9)\): The pairs (3,6), (4,5), (5,4), (6,3) result in the sum of 9. So, the probability is \( \frac{4}{36} = \frac{1}{9} \)f. \(P(10)\): The pairs (4,6), (5,5), (6,4) result in the sum of 10. So, the probability is \( \frac{3}{36} = \frac{1}{12} \)g. \(P(11)\): The pairs (5,6), (6,5) result in the sum of 11. So, the probability is \( \frac{2}{36} = \frac{1}{18} \)h. \(P(12)\): The pair (6,6) results in the sum of 12. So, the probability is \( \frac{1}{36} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dice Probability
Rolling a pair of dice is a classic example when learning about probability. The act of rolling dice provides a straightforward way of understanding random events and their associated probabilities. Each die has six faces, numbered from 1 to 6. When you roll two dice, you can achieve a variety of outcomes, defined by the sum of the numbers that land face-up. The key to understanding dice probability is recognizing that not all sums are equally likely. Some sums can occur through multiple combinations of numbers. The probability of each sum can be calculated by understanding the total possible outcomes and the specific outcomes that lead to that sum. With two dice, there are 36 total combinations (6 sides on the first die multiplied by 6 sides on the second die). This simplicity makes dice an excellent learning tool for probability concepts.
Outcome Calculation
The process of outcome calculation involves determining all the possible ways a certain event can occur. When it comes to rolling two dice, each die operates independently, which means you multiply the number of outcomes for each die.For example, to calculate an outcome for a sum of 7, consider the pairs that can make this happen: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). There are 6 combinations, so the probability of rolling a sum of 7 is \( \frac{6}{36} = \frac{1}{6} \).To find an outcome, follow these simple steps:
  • List all possible pairs of numbers that can occur.
  • Identify which of these pairs sum to your number of interest.
  • Count the number of pairs and divide by the total number of outcomes.
Probability Distribution
A probability distribution describes how probabilities are assigned to each possible outcome of a random event. In the case of rolling two dice, the distribution helps visualize the likelihood of rolling each possible sum, from 2 to 12. The distribution is not uniform; some sums can be achieved more often than others. For instance, a sum of 7 is the most common outcome because it can result from more combinations than any other sum. To understand a probability distribution:
  • Calculate the probability for each possible sum.
  • Use these probabilities to compare which outcomes are more likely.
  • Visualize this by plotting the probabilities to see the distribution shape.
This helps in predicting outcomes and understanding patterns inherent in the rolling of dice.
Elementary Statistics
Elementary statistics provide the foundational tools to analyze and interpret data. When you roll dice, you're performing a simple experiment that can be analyzed statistically. Statistics come into play in this process by:
  • Allowing you to calculate the probability of certain sums.
  • Understanding the likelihood of various outcomes through relative frequencies.
  • Offering insights into more complex concepts like variance and expected value.
For dice rolls, statistics make it possible to predict and understand randomness in a controlled manner. You collect data by rolling the dice multiple times, calculate probabilities, and then use statistical concepts to infer and explain these outcomes.

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