Suppose you take a pass-fail test repeatedly. Let Sk be the event that you are successful in your kth try, and Fk be the event that you fail the test in your kth try. On your first try, you have a 50 percent chance of passing the test.

P(S1) = 1 − P(F1) = 1/2.

Assume that as you take the test more often, your chance of failing the test goes down. In particular,

P(Fk)=12⋅P(Fk−1), for k=2,3,4,⋯

However, the result of different exams are independent. Suppose you take the test repeatedly until you pass the test for the first time. Let X be the total number of tests you take, so Range(X)={1,2,3,⋯}.

Find P(X=1),P(X=2),P(X=3).

Find a general formula for P(X=k) for k=1,2,⋯.

Find the probability that you take the test more than 2 times.

Given that you take the test more than once, find the probability that you take the test exactly twice.

After a dreary day of rain, the sun peeks through the clouds and a rainbow forms. you notice the rainbow is the shape of a parabola. the equation for this parabola is y = -x2 + 36. graph of a parabola opening down at the vertex 0 comma 36 crossing the x–axis at negative 6 comma 0 and 6 comma 0. in the distance, an airplane is taking off. as it ascends during take-off, it makes a slanted line that cuts through the rainbow at two points. create a table of at least four values for the function that includes two points of intersection between the airplane and the rainbow. analyze the two functions. answer the following reflection questions in complete sentences. what is the domain and range of the rainbow? explain what the domain and range represent. do all of the values make sense in this situation? why or why not? what are the x- and y-intercepts of the rainbow? explain what each intercept represents. is the linear function you created with your table positive or negative? explain. what are the solutions or solution to the system of equations created? explain what it or they represent. create your own piecewise function with at least two functions. explain, using complete sentences, the steps for graphing the function. graph the function by hand or using a graphing software of your choice (remember to submit the graph).