Question
your parents allow you to have an outing with your friends every Saturday of the month. you select the movies, bowling, skating, and the arcade as your activities for this month and put them on slips of paper in a jar to randomly select. In 3-5 sentences, described the independent and dependent probabilities, then determine how many possible events you could have for the month if the movie outing is selected first and not replaced.
Answers
GPT-4o mini
In this scenario, the independent probability refers to the chance of selecting any of the activities without any conditions based on previous selections. Each activity has an equal probability of being chosen during the first selection. The dependent probability comes into play when the first activity is selected and not replaced, meaning that the subsequent selections are affected by the previous choice, thereby changing the total number of available activities.
If you have four activities: movies, bowling, skating, and the arcade, and you select the movie first and do not replace it, you will have three activities remaining (bowling, skating, and arcade) for the remaining outings. Since you have four Saturdays to fill, you can choose from the remaining three activities for the second Saturday, two for the third Saturday, and one for the final Saturday. Thus, the total number of possible combinations for the month, after initially selecting the movie first, is calculated as:
\[
3! = 3 \times 2 \times 1 = 6 \text{ possible events}
\]
So, there are 6 different combinations of activities you could have for the month after selecting the movie first.
If you have four activities: movies, bowling, skating, and the arcade, and you select the movie first and do not replace it, you will have three activities remaining (bowling, skating, and arcade) for the remaining outings. Since you have four Saturdays to fill, you can choose from the remaining three activities for the second Saturday, two for the third Saturday, and one for the final Saturday. Thus, the total number of possible combinations for the month, after initially selecting the movie first, is calculated as:
\[
3! = 3 \times 2 \times 1 = 6 \text{ possible events}
\]
So, there are 6 different combinations of activities you could have for the month after selecting the movie first.