As a reward for your good grades, you are allowed to have an outing with your friends for the next 4 Saturdays. In order to choose what you are going to do, you made a list of activities: the movies, bowling, skating and the arcade. You put them on separate slips of paper and put them in a cup to randomly select.

Question

Question 1
What would you need to do as your are drawing events to determine if they are independent or dependent probabilities?(1 point)
0 / 10000 Word Limit
Question 2
If bowling is picked as the first activity, how many permutations are possible for the remaining events? List all the possibilites

Answer 1

Question 1

To determine if the probabilities of the events (activities) are independent or dependent, you need to perform the following steps:

1. Identify the Sample Space: First, list all the possible activities (movies, bowling, skating, and arcade).

2. Draw Activities Randomly: Each Saturday activity is drawn from the cup without replacing it, which means that the total number of activities decreases with each draw.

3. Calculate Probabilities: Calculate the probability of each activity being drawn at each step. For example, the probability of drawing bowling first is \( \frac{1}{4} \), and if bowling is drawn first, the probability of drawing any of the remaining activities (movies, skating, arcade) next changes to \( \frac{1}{3} \).

4. Assess Independence: Check whether the probability of drawing a specific activity (like bowling) affects the probability of drawing another specific activity (like skating) on the subsequent draw. If the probability of a second event is altered by the occurrence of the first event, the events are dependent. If they do not affect each other, they are independent.

In summary, you would need to draw the events, calculate their probabilities, and observe how they change with each draw to determine their independence or dependence.

Question 2

If bowling is picked as the first activity, the remaining activities are movies, skating, and the arcade. To determine the number of permutations of the remaining events, we can calculate the number of ways to arrange the three remaining activities.

The number of permutations of \( n \) activities is given by \( n! \) (n factorial). In this case:

• Remaining activities: Movies, Skating, Arcade (3 activities)

The number of permutations is: \[ 3! = 3 \times 2 \times 1 = 6 \]

Possible Permutations of Remaining Activities:

1. Movies, Skating, Arcade
2. Movies, Arcade, Skating
3. Skating, Movies, Arcade
4. Skating, Arcade, Movies
5. Arcade, Movies, Skating
6. Arcade, Skating, Movies

To summarize, if bowling is the first activity chosen, the permutations possible for the remaining events are 6 in total and they are listed above.