To determine the probability that Lorenzo purchased his ticket at the box office given that he paid more than [tex]$30, we'll analyze the given conditional relative frequency table.
The table presents the data as:
- For tickets costing "No More than $[/tex]30":
- The probability of purchasing online is 0.70.
- The probability of purchasing at the box office is 0.30.
- For tickets costing "More than [tex]$30":
- The probability of purchasing online is 0.86.
- The probability of purchasing at the box office is 0.14.
- The total probabilities across all costs are:
- The probability of purchasing online is 0.82.
- The probability of purchasing at the box office is 0.18.
We are asked to determine the probability that Lorenzo purchased his ticket at the box office given that he paid more than $[/tex]30. This is a conditional probability problem where we need to find [tex]\( P(\text{Box Office} \mid \text{More than } \$30) \)[/tex].
From the table, it is clear:
- [tex]\( P(\text{Box Office} \mid \text{More than } \$30) = 0.14 \)[/tex]
Therefore, the probability that Lorenzo purchased his ticket at the box office given that he paid more than $30 is [tex]\( \boxed{0.14} \)[/tex].
So, the correct answer is [tex]\( 0.14 \)[/tex].
