Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Our platform provides a seamless experience for finding reliable answers from a knowledgeable network of professionals. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
Let's solve this problem step-by-step.
### Step 1: Calculate Total Rolls
First, we need to calculate the total number of rolls. We can do this by summing up all the observed frequencies.
[tex]\[ \text{Total rolls} = 18 + 16 + 24 + 26 + 16 + 20 = 120 \][/tex]
### Step 2: Calculate the Experimental Probability of Rolling a 3
The experimental probability of rolling a number is the observed frequency of that number divided by the total number of rolls.
So, the experimental probability of rolling a 3 is:
[tex]\[ P(3)_{\text{experimental}} = \frac{\text{Observed frequency of 3}}{\text{Total rolls}} = \frac{24}{120} = \frac{1}{5} \][/tex]
### Step 3: Calculate the Theoretical Probability of Rolling a 3
The theoretical probability of rolling any specific number on a fair cube (die) is always:
[tex]\[ P(3)_{\text{theoretical}} = \frac{1}{6} \][/tex]
### Step 4: Compare the Experimental and Theoretical Probabilities of Rolling a 3
Next, we compare the experimental probability with the theoretical probability by finding the difference.
[tex]\[ \text{Difference} = P(3)_{\text{experimental}} - P(3)_{\text{theoretical}} \][/tex]
Substituting the values:
[tex]\[ \text{Difference} = \frac{1}{5} - \frac{1}{6} \][/tex]
To find a common denominator:
[tex]\[ \frac{1}{5} = \frac{6}{30} \][/tex]
[tex]\[ \frac{1}{6} = \frac{5}{30} \][/tex]
So,
[tex]\[ \text{Difference} = \frac{6}{30} - \frac{5}{30} = \frac{1}{30} \][/tex]
This means the experimental probability of rolling a 3 is [tex]\(\frac{1}{30}\)[/tex] greater than the theoretical probability of rolling a 3.
### Step 5: Calculate the Experimental Probability of Rolling a 2
Similarly, we calculate the experimental probability of rolling a 2.
[tex]\[ P(2)_{\text{experimental}} = \frac{16}{120} = \frac{2}{15} \][/tex]
### Step 6: Determine how the Experimental Probability of Rolling a 2 Compares to the Theoretical Probability of Rolling a 3
We have the theoretical probability of rolling a 3, which is:
[tex]\[ P(3)_{\text{theoretical}} = \frac{1}{6} \][/tex]
We need to compare this theoretical probability with the experimental probability of rolling a 2 by finding the difference.
[tex]\[ \text{Difference} = P(2)_{\text{experimental}} - P(3)_{\text{theoretical}} \][/tex]
Substituting the values:
[tex]\[ \text{Difference} = \frac{2}{15} - \frac{1}{6} \][/tex]
To find a common denominator:
[tex]\[ \frac{2}{15} = \frac{4}{30} \][/tex]
[tex]\[ \frac{1}{6} = \frac{5}{30} \][/tex]
So,
[tex]\[ \text{Difference} = \frac{4}{30} - \frac{5}{30} = -\frac{1}{30} \][/tex]
This means the experimental probability of rolling a 2 is [tex]\(\frac{1}{30}\)[/tex] less than the theoretical probability of rolling a 3.
### Summary
1. The experimental probability of rolling a 3 is [tex]\(\frac{1}{30}\)[/tex] greater than the theoretical probability of rolling a 3.
2. The experimental probability of rolling a 2 is [tex]\(\frac{1}{30}\)[/tex] less than the theoretical probability of rolling a 3.
### Step 1: Calculate Total Rolls
First, we need to calculate the total number of rolls. We can do this by summing up all the observed frequencies.
[tex]\[ \text{Total rolls} = 18 + 16 + 24 + 26 + 16 + 20 = 120 \][/tex]
### Step 2: Calculate the Experimental Probability of Rolling a 3
The experimental probability of rolling a number is the observed frequency of that number divided by the total number of rolls.
So, the experimental probability of rolling a 3 is:
[tex]\[ P(3)_{\text{experimental}} = \frac{\text{Observed frequency of 3}}{\text{Total rolls}} = \frac{24}{120} = \frac{1}{5} \][/tex]
### Step 3: Calculate the Theoretical Probability of Rolling a 3
The theoretical probability of rolling any specific number on a fair cube (die) is always:
[tex]\[ P(3)_{\text{theoretical}} = \frac{1}{6} \][/tex]
### Step 4: Compare the Experimental and Theoretical Probabilities of Rolling a 3
Next, we compare the experimental probability with the theoretical probability by finding the difference.
[tex]\[ \text{Difference} = P(3)_{\text{experimental}} - P(3)_{\text{theoretical}} \][/tex]
Substituting the values:
[tex]\[ \text{Difference} = \frac{1}{5} - \frac{1}{6} \][/tex]
To find a common denominator:
[tex]\[ \frac{1}{5} = \frac{6}{30} \][/tex]
[tex]\[ \frac{1}{6} = \frac{5}{30} \][/tex]
So,
[tex]\[ \text{Difference} = \frac{6}{30} - \frac{5}{30} = \frac{1}{30} \][/tex]
This means the experimental probability of rolling a 3 is [tex]\(\frac{1}{30}\)[/tex] greater than the theoretical probability of rolling a 3.
### Step 5: Calculate the Experimental Probability of Rolling a 2
Similarly, we calculate the experimental probability of rolling a 2.
[tex]\[ P(2)_{\text{experimental}} = \frac{16}{120} = \frac{2}{15} \][/tex]
### Step 6: Determine how the Experimental Probability of Rolling a 2 Compares to the Theoretical Probability of Rolling a 3
We have the theoretical probability of rolling a 3, which is:
[tex]\[ P(3)_{\text{theoretical}} = \frac{1}{6} \][/tex]
We need to compare this theoretical probability with the experimental probability of rolling a 2 by finding the difference.
[tex]\[ \text{Difference} = P(2)_{\text{experimental}} - P(3)_{\text{theoretical}} \][/tex]
Substituting the values:
[tex]\[ \text{Difference} = \frac{2}{15} - \frac{1}{6} \][/tex]
To find a common denominator:
[tex]\[ \frac{2}{15} = \frac{4}{30} \][/tex]
[tex]\[ \frac{1}{6} = \frac{5}{30} \][/tex]
So,
[tex]\[ \text{Difference} = \frac{4}{30} - \frac{5}{30} = -\frac{1}{30} \][/tex]
This means the experimental probability of rolling a 2 is [tex]\(\frac{1}{30}\)[/tex] less than the theoretical probability of rolling a 3.
### Summary
1. The experimental probability of rolling a 3 is [tex]\(\frac{1}{30}\)[/tex] greater than the theoretical probability of rolling a 3.
2. The experimental probability of rolling a 2 is [tex]\(\frac{1}{30}\)[/tex] less than the theoretical probability of rolling a 3.
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
