Let's analyze the given data and solve the problem step-by-step.
The table provided shows the following data:
- The probability of a customer having a shopping list and buying more items than expected is [tex]\( P(\text{More items} \cap \text{List}) = 0.17 \)[/tex].
- The probability of a customer not having a shopping list and buying more items than expected is [tex]\( P(\text{More items} \cap \text{No list}) = 0.18 \)[/tex].
- The probability of a customer buying more items than expected overall is [tex]\( P(\text{More items}) = 0.35 \)[/tex].
We are asked to find the probability that the customer did not use a shopping list given that they bought more items than expected. This is a conditional probability and can be found using the formula for conditional probability:
[tex]\[ P(\text{No list} \mid \text{More items}) = \frac{P(\text{More items} \cap \text{No list})}{P(\text{More items})} \][/tex]
Substituting the given values into the formula, we get:
[tex]\[ P(\text{No list} \mid \text{More items}) = \frac{0.18}{0.35} \][/tex]
Now, by evaluating this expression, we get:
[tex]\[ P(\text{No list} \mid \text{More items}) = 0.5142857142857143 \][/tex]
Rounded to two decimal places, the probability is:
[tex]\[ P(\text{No list} \mid \text{More items}) \approx 0.51 \][/tex]
Thus, if a customer purchased more items than planned, the likelihood that the customer did not use a shopping list is approximately [tex]\(0.51\)[/tex].
Therefore, the correct answer is [tex]\( \boxed{0.51} \)[/tex].
