Relative Frequency Worksheet | Printable PDF Worksheets - Free Printable

Problem Analysis and Solution

The task involves calculating relative frequencies and using them to answer questions about fairness, expected outcomes, and probabilities. Let's break it down section by section.

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## Section A: Relative Frequency Based on Experimental Data

Table 1: Dice Rolled 200 Times

#### Step 1: Calculate Relative Frequencies
The formula for relative frequency is:
\[
\text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total Number of Trials}}
\]
Here, the total number of trials is 200.

| N° | Frequency | Relative Frequency |
|----|-----------|--------------------|
| 1 | 20 | \( \frac{20}{200} = 0.10 \) |
| 2 | 30 | \( \frac{30}{200} = 0.15 \) |
| 3 | 25 | \( \frac{25}{200} = 0.125 \) |
| 4 | 20 | \( \frac{20}{200} = 0.10 \) |
| 5 | 25 | \( \frac{25}{200} = 0.125 \) |
| 6 | 80 | \( \frac{80}{200} = 0.40 \) |

#### Step 2: Answer the Questions
1. If the dice were rolled 1000 times, how many times would you expect to get a score of 6?
- The relative frequency of rolling a 6 is 0.40.
- Expected frequency in 1000 rolls:
\[
0.40 \times 1000 = 400
\]
- Answer: 400 times.

2. Do you think the dice is fair? Why?
- A fair six-sided die should have each number appear with a probability of \( \frac{1}{6} \approx 0.1667 \).
- Here, the relative frequencies are:
- 1: 0.10
- 2: 0.15
- 3: 0.125
- 4: 0.10
- 5: 0.125
- 6: 0.40
- The frequency of 6 (0.40) is significantly higher than expected for a fair die, while other numbers are lower. This suggests the die is not fair.
- Answer: No, the die is not fair because the relative frequency of 6 is much higher than expected.

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Table 2: Five-Sided Spinner Spun 120 Times

#### Step 1: Calculate Relative Frequencies
The total number of trials is 120.

| N° | Frequency | Relative Frequency |
|----|-----------|--------------------|
| 1 | 26 | \( \frac{26}{120} \approx 0.217 \) |
| 2 | 22 | \( \frac{22}{120} \approx 0.183 \) |
| 3 | 19 | \( \frac{19}{120} \approx 0.158 \) |
| 4 | 29 | \( \frac{29}{120} \approx 0.242 \) |
| 5 | 24 | \( \frac{24}{120} = 0.200 \) |

#### Step 2: Answer the Questions
3. If the spinner were spun 360 times, how many times would you expect to get a score of 2?
- The relative frequency of getting a 2 is approximately 0.183.
- Expected frequency in 360 spins:
\[
0.183 \times 360 \approx 66
\]
- Answer: 66 times.

4. Do you think the spinner is fair? Why?
- A fair five-sided spinner should have each number appear with a probability of \( \frac{1}{5} = 0.20 \).
- Here, the relative frequencies are:
- 1: 0.217
- 2: 0.183
- 3: 0.158
- 4: 0.242
- 5: 0.200
- While the frequencies are close to 0.20, there are slight deviations. However, these could be due to random variation in a small sample size (120 trials). Without more data, it is difficult to conclude definitively, but the spinner appears to be fair based on this data.
- Answer: Yes, the spinner appears to be fair because the relative frequencies are close to 0.20.

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## Section B: Relative Frequency Based on Survey Data

Table 1: Colour of Cars Passing Traffic

#### Step 1: Calculate Relative Frequencies
The total number of cars is:
\[
13 + 34 + 25 + 10 + 8 = 90
\]

| Colour | Frequency | Relative Frequency |
|--------|-----------|--------------------|
| Red | 13 | \( \frac{13}{90} \approx 0.144 \) |
| Silver | 34 | \( \frac{34}{90} \approx 0.378 \) |
| Black | 25 | \( \frac{25}{90} \approx 0.278 \) |
| White | 10 | \( \frac{10}{90} \approx 0.111 \) |
| Blue | 8 | \( \frac{8}{90} \approx 0.089 \) |

#### Step 2: Answer the Questions
1. What is the relative frequency of the next passing car being silver?
- The relative frequency of silver cars is approximately 0.378.
- Answer: 0.378.

2. Estimate the number of white cars you would expect to see if 4000 cars passed by?
- The relative frequency of white cars is approximately 0.111.
- Expected number of white cars in 4000:
\[
0.111 \times 4000 \approx 444
\]
- Answer: 444 cars.

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Table 2: Counter Taken from a Bag

#### Step 1: Calculate Relative Frequencies
The relative frequency is calculated as:
\[
\text{Relative Frequency} = \frac{\text{Frequency of Blue Counters}}{\text{Number of Trials}}
\]

| Number of Trials | Frequency | Relative Frequency |
|------------------|-----------|--------------------|
| 50 | 8 | \( \frac{8}{50} = 0.160 \) |
| 100 | 22 | \( \frac{22}{100} = 0.220 \) |
| 150 | 48 | \( \frac{48}{150} = 0.320 \) |
| 200 | 59 | \( \frac{59}{200} = 0.295 \) |

#### Step 2: Answer the Questions
3. What is the best estimate of the probability of picking a blue counter?
- The relative frequency stabilizes around 0.295 (from 200 trials).
- Answer: 0.295.

4. Estimate the probability of taking a blue counter from the bag.
- Based on the data, the probability of picking a blue counter is approximately 0.295.
- Answer: 0.295.

5. How could you get a more accurate estimate of the probability of taking out a blue counter?
- To get a more accurate estimate, increase the number of trials. The law of large numbers suggests that as the number of trials increases, the relative frequency will converge closer to the true probability.
- Answer: Increase the number of trials.

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## Final Answers
1. Dice Rolled 1000 Times (Score of 6): \(\boxed{400}\)
2. Fairness of Dice: No, the die is not fair.
3. Spinner Spun 360 Times (Score of 2): \(\boxed{66}\)
4. Fairness of Spinner: Yes, the spinner appears to be fair.
5. Relative Frequency of Silver Car: \(\boxed{0.378}\)
6. Expected White Cars in 4000: \(\boxed{444}\)
7. Best Estimate of Probability of Blue Counter: \(\boxed{0.295}\)
8. Probability of Taking a Blue Counter: \(\boxed{0.295}\)
9. How to Improve Accuracy: Increase the number of trials.