Answer :
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.
