Question

A burrito restaurant recorded 200 customer orders. Each order was categorized by whether it was takeout or dine-in and whether it was placed on a weekday or a weekend.

The results are shown below:





Using this table, decide which statement about independence is true. Which statement is correct?

The events “takeout” and “weekday” are independent.
The events “weekday” and “weekend” are independent.
The events “order type” (takeout or dine-in) and “day of the week” (weekday or weekend) are not independent.
The events “takeout” and “dine-in” are independent.

Answer 1

Answer: "The events 'order type' (takeout or dine-in) and 'day of the week' (weekday or weekend) are not independent."

Reason: independence requires P(A and B) = P(A)P(B). The pairs "weekday" and "weekend" and "takeout" and "dine-in" are complements, so they cannot be independent unless one of them has probability 0 or 1 (because P(weekday and weekend) = 0 but P(weekday)P(weekend) > 0 for nontrivial data). That eliminates the second and fourth choices.

That leaves checking whether takeout and weekday are independent. Using the table you test whether P(takeout and weekday) = P(takeout)P(weekday) (or equivalently whether P(takeout | weekday) = P(takeout)). The table’s counts do not satisfy that equality, so takeout and weekday are not independent. Therefore the correct statement is that order type and day of week are not independent.