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A pair of fair dice are rolled one time. What is the probability that the sum of the two dice is equal to 4 or the sum is an odd number?

A pair of fair dice are rolled one time. What is the probability that the sum of the two dice is equal to 4 or the sum is an odd number?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Transcription for Educational Website:** --- **Title: Understanding Probability with Dice** **Question:** A pair of fair dice are rolled one time. What is the probability that the sum of the two dice is equal to 4 or the sum is an odd number? --- **Explanation:** In this problem, we are asked to determine the probability of two specific outcomes when rolling two dice: 1. The sum of the two dice equals 4. 2. The sum of the two dice is an odd number. To solve this, we consider the possible outcomes from rolling two six-sided dice. Each die has faces numbered from 1 to 6, giving us a total of 6x6 = 36 combinations. **Outcome 1: Sum equals 4** The possible combinations that result in a sum of 4 are: - (1, 3) - (2, 2) - (3, 1) There are 3 outcomes where the sum is equal to 4. **Outcome 2: Sum is odd** To find the number of outcomes where the sum is odd, we list out the combinations for each possible sum that is odd (3, 5, 7, 9, and 11): - Sum = 3: (1, 2), (2, 1) - Sum = 5: (1, 4), (2, 3), (3, 2), (4, 1) - Sum = 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) - Sum = 9: (3, 6), (4, 5), (5, 4), (6, 3) - Sum = 11: (5, 6), (6, 5) In total, there are 18 outcomes where the sum is odd. **Combining the Probabilities:** Since one of the outcomes (sum equals 4) is even, we don't double-count it. Therefore, the probability is calculated by observing that there are unique combinations in each case. Total unique possible outcomes for our condition: - Sum equals 4: 3 outcomes - Sum is odd: 18 outcomes However, we do not double-count (1, 3) and (3, 1). Since the
Transcribed Image Text:**Transcription for Educational Website:** --- **Title: Understanding Probability with Dice** **Question:** A pair of fair dice are rolled one time. What is the probability that the sum of the two dice is equal to 4 or the sum is an odd number? --- **Explanation:** In this problem, we are asked to determine the probability of two specific outcomes when rolling two dice: 1. The sum of the two dice equals 4. 2. The sum of the two dice is an odd number. To solve this, we consider the possible outcomes from rolling two six-sided dice. Each die has faces numbered from 1 to 6, giving us a total of 6x6 = 36 combinations. **Outcome 1: Sum equals 4** The possible combinations that result in a sum of 4 are: - (1, 3) - (2, 2) - (3, 1) There are 3 outcomes where the sum is equal to 4. **Outcome 2: Sum is odd** To find the number of outcomes where the sum is odd, we list out the combinations for each possible sum that is odd (3, 5, 7, 9, and 11): - Sum = 3: (1, 2), (2, 1) - Sum = 5: (1, 4), (2, 3), (3, 2), (4, 1) - Sum = 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) - Sum = 9: (3, 6), (4, 5), (5, 4), (6, 3) - Sum = 11: (5, 6), (6, 5) In total, there are 18 outcomes where the sum is odd. **Combining the Probabilities:** Since one of the outcomes (sum equals 4) is even, we don't double-count it. Therefore, the probability is calculated by observing that there are unique combinations in each case. Total unique possible outcomes for our condition: - Sum equals 4: 3 outcomes - Sum is odd: 18 outcomes However, we do not double-count (1, 3) and (3, 1). Since the
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