Which of the following are independent events?
a. knowing that you have to get up early tomorrow, and going to bed before 9 p. m.
b. knowing that you have a test in school tomorrow, and studying the night before
c. knowing that it is going to rain tomorrow, and bringing an umbrella to school
d. knowing that you have a test in school tomorrow, and bringing an umbrella to school

Option D

Step-by-step explanation:

To show : which of the statements is independent event

By definition of independent events, if there are two things occurred then the occurrence of first event is not dependent on occurrence of second event or vice versa.

if we notice in all statement occurrence of first event is dependent on occurrence of second event except the last one as knowing that you have a test in a school tomorrow and bringing an umbrella to school have no such dependency or need.

Therefore Option D is independent event

when rotating a figure 180 degrees, imagine that you are able to take the figure and turn it clockwise or counterclockwise around a center point. to rotate a figure 180 degrees, you will need to apply the rule (x, y) → (-x, -y). start by using a coordinate grid with coordinates for each vertex of the figure. the center point of the coordinate grid is located at (0, 0), which is what you will rotate the figure around. write down the original coordinates of the shape you are going to rotate. then, apply the rule. for example, coordinate (1, 2) becomes (-1, -2). another example would be, coordinate (-4, -2) becoming (4, 2). after you change each original coordinate to the rotated coordinate, you will draw your new figure.

step-by-step explanation: