Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
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.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
