Answer :
To solve this problem, we need to calculate both the theoretical and experimental probabilities of drawing a blue tile. Let's take this step by step:
### Step 1: Calculate the Total Number of Draws and Blues
First, let's sum up the number of draws and blues across all trials:
- Total number of draws:
[tex]\[ 54 + 36 + 80 + 22 + 75 = 267 \][/tex]
- Total number of blues:
[tex]\[ 18 + 11 + 20 + 4 + 32 = 85 \][/tex]
### Step 2: Calculate the Experimental Probability
The experimental probability of drawing a blue tile is the ratio of the total number of blue draws to the total number of draws, multiplied by 100 to convert it to a percentage.
[tex]\[ \text{Experimental Probability} = \left( \frac{85}{267} \right) \times 100 \approx 31.84\% \][/tex]
### Step 3: Calculate the Theoretical Probability for Each Trial
For each trial, calculate the probability of drawing a blue tile, then find the average:
- Trial 1:
[tex]\[ \left(\frac{18}{54}\right) \times 100 \approx 33.33\%\][/tex]
- Trial 2:
[tex]\[ \left(\frac{11}{36}\right) \times 100 \approx 30.56\% \][/tex]
- Trial 3:
[tex]\[ \left(\frac{20}{80}\right) \times 100 \approx 25.00\% \][/tex]
- Trial 4:
[tex]\[ \left(\frac{4}{22}\right) \times 100 \approx 18.18\% \][/tex]
- Trial 5:
[tex]\[ \left(\frac{32}{75}\right) \times 100 \approx 42.67\% \][/tex]
Now, average these probabilities:
[tex]\[ \text{Theoretical Probability} = \left( \frac{33.33 + 30.56 + 25.00 + 18.18 + 42.67}{5} \right) \approx 29.95\% \][/tex]
### Step 4: Determine the Difference
Finally, we compare the experimental and theoretical probabilities:
- Difference:
[tex]\[ \text{Difference} = 31.84\% - 29.95\% \approx 1.89 \text{ percentage points} \][/tex]
### Conclusion
The experimental probability is greater than the theoretical probability by approximately 1.89 percentage points.
Given the options:
a. The theoretical probability is 2.78 percentage points greater than the experimental probability.
b. The theoretical probability is 1.49 percentage points greater than the experimental probability.
c. The experimental probability is 4.06 percentage points greater than the theoretical probability.
d. The experimental probability is 2.17 percentage points greater than the theoretical probability.
None of these options are correct. The correct answer should reflect the difference as 1.89 percentage points.
### Step 1: Calculate the Total Number of Draws and Blues
First, let's sum up the number of draws and blues across all trials:
- Total number of draws:
[tex]\[ 54 + 36 + 80 + 22 + 75 = 267 \][/tex]
- Total number of blues:
[tex]\[ 18 + 11 + 20 + 4 + 32 = 85 \][/tex]
### Step 2: Calculate the Experimental Probability
The experimental probability of drawing a blue tile is the ratio of the total number of blue draws to the total number of draws, multiplied by 100 to convert it to a percentage.
[tex]\[ \text{Experimental Probability} = \left( \frac{85}{267} \right) \times 100 \approx 31.84\% \][/tex]
### Step 3: Calculate the Theoretical Probability for Each Trial
For each trial, calculate the probability of drawing a blue tile, then find the average:
- Trial 1:
[tex]\[ \left(\frac{18}{54}\right) \times 100 \approx 33.33\%\][/tex]
- Trial 2:
[tex]\[ \left(\frac{11}{36}\right) \times 100 \approx 30.56\% \][/tex]
- Trial 3:
[tex]\[ \left(\frac{20}{80}\right) \times 100 \approx 25.00\% \][/tex]
- Trial 4:
[tex]\[ \left(\frac{4}{22}\right) \times 100 \approx 18.18\% \][/tex]
- Trial 5:
[tex]\[ \left(\frac{32}{75}\right) \times 100 \approx 42.67\% \][/tex]
Now, average these probabilities:
[tex]\[ \text{Theoretical Probability} = \left( \frac{33.33 + 30.56 + 25.00 + 18.18 + 42.67}{5} \right) \approx 29.95\% \][/tex]
### Step 4: Determine the Difference
Finally, we compare the experimental and theoretical probabilities:
- Difference:
[tex]\[ \text{Difference} = 31.84\% - 29.95\% \approx 1.89 \text{ percentage points} \][/tex]
### Conclusion
The experimental probability is greater than the theoretical probability by approximately 1.89 percentage points.
Given the options:
a. The theoretical probability is 2.78 percentage points greater than the experimental probability.
b. The theoretical probability is 1.49 percentage points greater than the experimental probability.
c. The experimental probability is 4.06 percentage points greater than the theoretical probability.
d. The experimental probability is 2.17 percentage points greater than the theoretical probability.
None of these options are correct. The correct answer should reflect the difference as 1.89 percentage points.
