To find the probability that the service was good given that the meal was lunch, we need to use the concept of conditional probability.
The probability of an event [tex]\( A \)[/tex] occurring given that event [tex]\( B \)[/tex] has occurred can be calculated using the formula:
[tex]\[
P(A|B) = \frac{P(A \cap B)}{P(B)}
\][/tex]
In this problem, we want to find the probability that the service was good (event [tex]\( A \)[/tex]) given that the meal was lunch (event [tex]\( B \)[/tex]).
From the table, we observe the following:
- The number of times the service was good during lunch ([tex]\( A \cap B \)[/tex]) = 22.
- The total number of lunch meals ([tex]\( B \)[/tex]) = 72.
So, we calculate the probability as:
[tex]\[
P(\text{Service good | Lunch}) = \frac{\text{Number of times service was good during lunch}}{\text{Total number of lunch meals}} = \frac{22}{72}
\][/tex]
Simplifying the fraction:
[tex]\[
\frac{22}{72} = \frac{11}{36}
\][/tex]
Thus, the probability that the service was good given that the meal was lunch is:
[tex]\[
\boxed{\frac{11}{36}}
\][/tex]
