Question

Tyler believes that an 8-sided die can be used to predict whether customers at his store will use a coupon when making their purchases, with a roll of 1 indicating a coupon and all other outcomes indicating no coupon. Today, 4 of the first 48 customers in the store used a coupon. How does the experimental probability of a customer using a coupon compare to the theoretical probability of Tyler’s die?
The experimental probability is greater than the theoretical probability by more than 2%.
The experimental probability is less than the theoretical probability by more than 2%.
The experimental probability is greater than the theoretical probability, but within 2%.
The experimental probability is less than the theoretical probability, but within 2%.

Answer 1

To determine how the experimental probability compares to the theoretical probability, we first need to calculate both probabilities.

1. **Theoretical Probability**:
Tyler believes that a roll of a single 8-sided die indicating a coupon is represented by a roll of 1. The theoretical probability of using a coupon (rolling a 1) is:

\[
P(\text{coupon}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{8} = 0.125 \text{ or } 12.5\%
\]

2. **Experimental Probability**:
From the data provided, 4 out of 48 customers used a coupon. The experimental probability of a customer using a coupon is calculated as follows:

\[
P(\text{coupon, experimental}) = \frac{\text{Number of customers who used a coupon}}{\text{Total number of customers}} = \frac{4}{48} = \frac{1}{12} \approx 0.0833 \text{ or } 8.33\%
\]

3. **Comparison**:
- Theoretical Probability: 12.5%
- Experimental Probability: 8.33%

Next, we calculate the difference between the two probabilities:

\[
\text{Difference} = P(\text{coupon, theoretical}) - P(\text{coupon, experimental}) = 12.5\% - 8.33\% \approx 4.17\%
\]

Now, let's analyze the options based on the calculated probabilities:

- The experimental probability of 8.33% is less than the theoretical probability of 12.5% by approximately 4.17%, which is greater than 2%.

Thus, the correct conclusion is:

**The experimental probability is less than the theoretical probability by more than 2%.**