Two Dice Are Thrown Simultaneously. Find The Probability That:a. Both Dice Have The Same Number.b. The Total Number On The Dice Is Greater Than 9.c. The Sum Of Numbers On The Dice Is 9.d. Both Numbers On The Dice Are Prime Numbers.
Introduction
Probability is a fundamental concept in mathematics that deals with the likelihood of an event occurring. In this article, we will explore the probability of various events related to the roll of two dice. We will use the basic principles of probability to calculate the probabilities of these events and provide a detailed analysis of each scenario.
Event a: Both Dice Have the Same Number
When two dice are thrown simultaneously, there are 36 possible outcomes, as each die has 6 possible outcomes (1, 2, 3, 4, 5, 6). To find the probability that both dice have the same number, we need to count the number of favorable outcomes.
| Die 1 | Die 2 | Outcome |
|---|---|---|
| 1 | 1 | (1,1) |
| 2 | 2 | (2,2) |
| 3 | 3 | (3,3) |
| 4 | 4 | (4,4) |
| 5 | 5 | (5,5) |
| 6 | 6 | (6,6) |
There are 6 favorable outcomes where both dice have the same number. Therefore, the probability of this event is:
P(Both dice have the same number) = Number of favorable outcomes / Total number of outcomes = 6 / 36 = 1/6
Event b: The Total Number on the Dice is Greater than 9
To find the probability that the total number on the dice is greater than 9, we need to count the number of favorable outcomes.
| Die 1 | Die 2 | Outcome | Total |
|---|---|---|---|
| 1 | 1 | (1,1) | 2 |
| 2 | 2 | (2,2) | 4 |
| 3 | 3 | (3,3) | 6 |
| 4 | 4 | (4,4) | 8 |
| 5 | 5 | (5,5) | 10 |
| 6 | 6 | (6,6) | 12 |
| 5 | 6 | (5,6) | 11 |
| 6 | 5 | (6,5) | 11 |
| 6 | 6 | (6,6) | 12 |
There are 10 favorable outcomes where the total number on the dice is greater than 9. Therefore, the probability of this event is:
P(The total number on the dice is greater than 9) = Number of favorable outcomes / Total number of outcomes = 10 / 36 = 5/18
Event c: The Sum of Numbers on the Dice is 9
To find the probability that the sum of numbers on the dice is 9, we need to count the number of favorable outcomes.
| Die 1 | Die 2 | Outcome | Sum |
|---|---|---|---|
| 3 | 6 | (3,6) | 9 |
| 6 | 3 | (6,3) | 9 |
| 4 | 5 | (4,5) | 9 |
| 5 | 4 | (5,4) | 9 |
There are 4 favorable outcomes where the sum of numbers on the dice is 9. Therefore, the probability of this event is:
P(The sum of numbers on the dice is 9) = Number of favorable outcomes / Total number of outcomes = 4 / 36 = 1/9
Event d: Both Numbers on the Dice are Prime Numbers
To find the probability that both numbers on the dice are prime numbers, we need to count the number of favorable outcomes.
| Die 1 | Die 2 | Outcome | Prime |
|---|---|---|---|
| 2 | 3 | (2,3) | Yes |
| 3 | 2 | (3,2) | Yes |
| 2 | 2 | (2,2) | No |
| 3 | 3 | (3,3) | No |
There are 2 favorable outcomes where both numbers on the dice are prime numbers. Therefore, the probability of this event is:
P(Both numbers on the dice are prime numbers) = Number of favorable outcomes / Total number of outcomes = 2 / 36 = 1/18
Conclusion
In this article, we have explored the probability of various events related to the roll of two dice. We have used the basic principles of probability to calculate the probabilities of these events and provided a detailed analysis of each scenario. The probabilities of the events are:
- P(Both dice have the same number) = 1/6
- P(The total number on the dice is greater than 9) = 5/18
- P(The sum of numbers on the dice is 9) = 1/9
- P(Both numbers on the dice are prime numbers) = 1/18
Q&A: Probability of Dice Rolls
Q: What is the probability that both dice have the same number?
A: The probability that both dice have the same number is 1/6. This is because there are 6 favorable outcomes (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6) out of a total of 36 possible outcomes.
Q: What is the probability that the total number on the dice is greater than 9?
A: The probability that the total number on the dice is greater than 9 is 5/18. This is because there are 10 favorable outcomes (5,6), (6,5), (5,5), (6,6), (4,6), (6,4), (5,7), (7,5), (6,7), and (7,6) out of a total of 36 possible outcomes.
Q: What is the probability that the sum of numbers on the dice is 9?
A: The probability that the sum of numbers on the dice is 9 is 1/9. This is because there are 4 favorable outcomes (3,6), (6,3), (4,5), and (5,4) out of a total of 36 possible outcomes.
Q: What is the probability that both numbers on the dice are prime numbers?
A: The probability that both numbers on the dice are prime numbers is 1/18. This is because there are 2 favorable outcomes (2,3) and (3,2) out of a total of 36 possible outcomes.
Q: What is the probability that the first die is a 6 and the second die is a 5?
A: The probability that the first die is a 6 and the second die is a 5 is 1/36. This is because there is only 1 favorable outcome (6,5) out of a total of 36 possible outcomes.
Q: What is the probability that the first die is not a 6 and the second die is not a 5?
A: The probability that the first die is not a 6 and the second die is not a 5 is 25/36. This is because there are 25 favorable outcomes (1,1), (1,2), (1,3), (1,4), (1,5), (2,1), (2,2), (2,3), (2,4), (2,5), (3,1), (3,2), (3,3), (3,4), (3,5), (4,1), (4,2), (4,3), (4,4), (4,5), (5,1), (5,2), (5,3), (5,4), and (5,5) out of a total of 36 possible outcomes.
Q: What is the probability that the first die is greater than the second die?
A: The probability that the first die is greater than the second die is 15/36. This is because there are 15 favorable outcomes (1,1), (1,2), (1,3), (1,4), (1,5), (2,1), (2,2), (2,3), (2,4), (2,5), (3,1), (3,2), (3,3), (3,4), and (3,5) out of a total of 36 possible outcomes.
Q: What is the probability that the first die is less than the second die?
A: The probability that the first die is less than the second die is 15/36. This is because there are 15 favorable outcomes (1,1), (2,1), (3,1), (4,1), (5,1), (1,2), (1,3), (1,4), (1,5), (2,2), (2,3), (2,4), (2,5), (3,3), (3,4), (3,5), (4,4), (4,5), (5,5), and (6,6) out of a total of 36 possible outcomes.
Conclusion
In this article, we have provided a comprehensive analysis of the probability of various events related to the roll of two dice. We have used the basic principles of probability to calculate the probabilities of these events and provided a detailed analysis of each scenario. The probabilities of the events are:
- P(Both dice have the same number) = 1/6
- P(The total number on the dice is greater than 9) = 5/18
- P(The sum of numbers on the dice is 9) = 1/9
- P(Both numbers on the dice are prime numbers) = 1/18
- P(The first die is a 6 and the second die is a 5) = 1/36
- P(The first die is not a 6 and the second die is not a 5) = 25/36
- P(The first die is greater than the second die) = 15/36
- P(The first die is less than the second die) = 15/36
These results demonstrate the importance of probability in understanding the likelihood of events in various scenarios.