Certainly! Let's work through this problem step-by-step.
### Step 1: Understanding the Data
We are given the outcomes of rolling a six-sided die, along with the frequency of each outcome:
| Outcome Rolled | 1 | 2 | 3 | 4 | 5 | 6 |
|----------------|---|----|----|----|----|----|
| Number of Rolls | 91 | 70 | 80 | 68 | 78 | 65 |
### Step 2: Calculate Total Number of Rolls
To find the total number of rolls, we need to sum the frequencies:
Total Rolls = 91 + 70 + 80 + 68 + 78 + 65 = 452
### Step 3: Calculate the Probability of Each Outcome
The probability of each outcome is calculated by dividing the frequency of that outcome by the total number of rolls. Here are the probabilities for each outcome:
- Probability of rolling a 1: [tex]\( \frac{91}{452} \approx 0.201 \)[/tex]
- Probability of rolling a 2: [tex]\( \frac{70}{452} \approx 0.155 \)[/tex]
- Probability of rolling a 3: [tex]\( \frac{80}{452} \approx 0.177 \)[/tex]
- Probability of rolling a 4: [tex]\( \frac{68}{452} \approx 0.150 \)[/tex]
- Probability of rolling a 5: [tex]\( \frac{78}{452} \approx 0.173 \)[/tex]
- Probability of rolling a 6: [tex]\( \frac{65}{452} \approx 0.144 \)[/tex]
### Step 4: Convert Probabilities to the Nearest Thousandth
We can summarize the probabilities rounded to the nearest thousandth as follows:
| Outcome Rolled | 1 | 2 | 3 | 4 | 5 | 6 |
|----------------|-------|-------|-------|------|------|------|
| Probability | 0.201 | 0.155 | 0.177 | 0.150| 0.173| 0.144|
### Summary
- Total Number of Rolls: 452
- Probabilities, rounded to the nearest thousandth:
- Probability of rolling a 1: 0.201
- Probability of rolling a 2: 0.155
- Probability of rolling a 3: 0.177
- Probability of rolling a 4: 0.150
- Probability of rolling a 5: 0.173
- Probability of rolling a 6: 0.144
I hope this detailed step-wise explanation helps you in understanding how to calculate the total number of rolls and the probability of each outcome, based on the given frequencies.
