In this scenario, the independent probability refers to the selection of the first activity (e.g., movies), which doesn't rely on any other activity being chosen beforehand. The dependent probability comes into play when considering the subsequent activities, since the selection of one option affects the available choices for the remaining outings. If you have four activities (movies, bowling, skating, and the arcade) and you select movies first without replacement, then you are left with three remaining activities to choose from for the next outings.
Since there are a total of 4 activities and if you select movies first, the number of possible events for the month with the remaining three activities is calculated by finding the permutations of the remaining activities. So you would have 3! (3 factorial), which equals 6 different ways to arrange the remaining activities. Thus, the total number of possible events for the month would be \(6\) arrangements after the first selection of movies.