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13. You roll two dice, one red and one green. Losing combina- tions are doubles (both dice show the same number) and out- comes in which the green die shows an odd number and the red die shows an even number. The other combinations are winning ones. How many winning combinations are there?

13. You roll two dice, one red and one green. Losing combina- tions are doubles (both dice show the same number) and out- comes in which the green die shows an odd number and the red die shows an even number. The other combinations are winning ones. How many winning combinations are there?

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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**Exercise 13** **Problem Statement:** You roll two dice, one red and one green. Losing combinations are doubles (both dice show the same number) and outcomes in which the green die shows an odd number and the red die shows an even number. The other combinations are winning ones. How many winning combinations are there? **Solution:** To solve this problem, we consider the total number of possible outcomes and subtract the losing combinations. **Total Possible Outcomes:** - Each die has 6 faces. - Total combinations when rolling two dice = \(6 \times 6 = 36\). **Losing Combinations:** 1. **Doubles:** - These are combinations where both dice show the same number: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6). - Total number of doubles = 6. 2. **Green Die Shows Odd and Red Die Shows Even:** - Green die odd numbers: 1, 3, 5. - Red die even numbers: 2, 4, 6. - Number of combinations = \(3 \times 3 = 9\). **Total Losing Combinations:** - Doubles + Green odd/Red even = 6 + 9 = 15. **Winning Combinations:** - Total winning combinations = Total outcomes - Losing combinations = 36 - 15 = 21. Thus, there are 21 winning combinations.
Transcribed Image Text:**Exercise 13** **Problem Statement:** You roll two dice, one red and one green. Losing combinations are doubles (both dice show the same number) and outcomes in which the green die shows an odd number and the red die shows an even number. The other combinations are winning ones. How many winning combinations are there? **Solution:** To solve this problem, we consider the total number of possible outcomes and subtract the losing combinations. **Total Possible Outcomes:** - Each die has 6 faces. - Total combinations when rolling two dice = \(6 \times 6 = 36\). **Losing Combinations:** 1. **Doubles:** - These are combinations where both dice show the same number: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6). - Total number of doubles = 6. 2. **Green Die Shows Odd and Red Die Shows Even:** - Green die odd numbers: 1, 3, 5. - Red die even numbers: 2, 4, 6. - Number of combinations = \(3 \times 3 = 9\). **Total Losing Combinations:** - Doubles + Green odd/Red even = 6 + 9 = 15. **Winning Combinations:** - Total winning combinations = Total outcomes - Losing combinations = 36 - 15 = 21. Thus, there are 21 winning combinations.
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