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Sagot :
Absolutely! Let's tackle the problem step by step.
### Step 1: Identify the Range of Possible Sums
When two six-sided dice are rolled, the smallest sum we can get is [tex]\(1 + 1 = 2\)[/tex] and the largest sum is [tex]\(6 + 6 = 12\)[/tex]. So, the possible sums range from 2 to 12.
### Step 2: Determine Primes between 2 and 12
First, identify which numbers in this range are prime.
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The prime numbers between 2 and 12 are:
- 2
- 3
- 5
- 7
- 11
### Step 3: Count the Total Possible Outcomes
Each die has 6 faces. When two dice are rolled, the total number of possible outcomes is:
[tex]\[ 6 \times 6 = 36 \][/tex]
### Step 4: Identify Favorable Outcomes
Now, let's determine the number of outcomes where the sum of the two dice is a prime number. We consider each sum and see if it is prime.
- Sum of 2: [tex]\( (1,1) \)[/tex]
- Sum of 3: [tex]\( (1,2), (2,1) \)[/tex]
- Sum of 5: [tex]\( (1,4), (2,3), (3,2), (4,1) \)[/tex]
- Sum of 7: [tex]\( (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) \)[/tex]
- Sum of 11: [tex]\( (5,6), (6,5) \)[/tex]
Now, let's count these outcomes:
- Sum of 2: 1 way
- Sum of 3: 2 ways
- Sum of 5: 4 ways
- Sum of 7: 6 ways
- Sum of 11: 2 ways
Adding these up:
[tex]\[ 1 + 2 + 4 + 6 + 2 = 15 \][/tex]
So, there are 15 favorable outcomes where the sum of the two dice is a prime number.
### Step 5: Calculate the Probability
Finally, the probability of this event happening is the ratio of the number of favorable outcomes to the total number of possible outcomes:
[tex]\[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} = \frac{15}{36} \][/tex]
Simplify the fraction (if possible):
[tex]\[ \frac{15}{36} = \frac{5}{12} \][/tex]
As a decimal, this is approximately:
[tex]\[ \frac{5}{12} \approx 0.4167 \][/tex]
So, the probability that the sum of the numbers on the two dice will be a prime number is:
[tex]\[ \boxed{0.4167} \][/tex]
In summary, the probability that the two numbers rolled on the dice add up to a prime number is approximately 0.4167, or [tex]\(\frac{5}{12}\)[/tex].
### Step 1: Identify the Range of Possible Sums
When two six-sided dice are rolled, the smallest sum we can get is [tex]\(1 + 1 = 2\)[/tex] and the largest sum is [tex]\(6 + 6 = 12\)[/tex]. So, the possible sums range from 2 to 12.
### Step 2: Determine Primes between 2 and 12
First, identify which numbers in this range are prime.
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The prime numbers between 2 and 12 are:
- 2
- 3
- 5
- 7
- 11
### Step 3: Count the Total Possible Outcomes
Each die has 6 faces. When two dice are rolled, the total number of possible outcomes is:
[tex]\[ 6 \times 6 = 36 \][/tex]
### Step 4: Identify Favorable Outcomes
Now, let's determine the number of outcomes where the sum of the two dice is a prime number. We consider each sum and see if it is prime.
- Sum of 2: [tex]\( (1,1) \)[/tex]
- Sum of 3: [tex]\( (1,2), (2,1) \)[/tex]
- Sum of 5: [tex]\( (1,4), (2,3), (3,2), (4,1) \)[/tex]
- Sum of 7: [tex]\( (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) \)[/tex]
- Sum of 11: [tex]\( (5,6), (6,5) \)[/tex]
Now, let's count these outcomes:
- Sum of 2: 1 way
- Sum of 3: 2 ways
- Sum of 5: 4 ways
- Sum of 7: 6 ways
- Sum of 11: 2 ways
Adding these up:
[tex]\[ 1 + 2 + 4 + 6 + 2 = 15 \][/tex]
So, there are 15 favorable outcomes where the sum of the two dice is a prime number.
### Step 5: Calculate the Probability
Finally, the probability of this event happening is the ratio of the number of favorable outcomes to the total number of possible outcomes:
[tex]\[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} = \frac{15}{36} \][/tex]
Simplify the fraction (if possible):
[tex]\[ \frac{15}{36} = \frac{5}{12} \][/tex]
As a decimal, this is approximately:
[tex]\[ \frac{5}{12} \approx 0.4167 \][/tex]
So, the probability that the sum of the numbers on the two dice will be a prime number is:
[tex]\[ \boxed{0.4167} \][/tex]
In summary, the probability that the two numbers rolled on the dice add up to a prime number is approximately 0.4167, or [tex]\(\frac{5}{12}\)[/tex].
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