In basketball, a one-and-one free throw situation sometimes occurs when a player is fouled. The fouled player gets to take a shot from the free-throw line, and if he makes the first shot then he gets a second shot. If he misses the first shot, that's the end and no points are scored. Each basket made is worth 1 point. So when it's all over, he will have scored 0, 1, or 2 points.

Scenario 1: Ann is a 82% free throw shooter. Let's assume that her free throw attempts are independent trials, i.e. every time she shoots there will always be a 0.82 probability that she makes the shot.

Hints and Tips: Draw a tree diagram of this situation, put in the numbers, and "multiply up" the branches to find the probabilities of each of the three outcomes. Use this tree and the concept of expected value to answer the following question. What is the expected value for the number of points Ann will score if she is fouled in a one-and-one situation? (Give your answer correct to three decimal places.)

Scenario 2: Isaac is a basketball player who averages scoring 1.4 points whenever he has a one-and-one free throw situation. In other words, the expected value for the number of points Isaac will score if he is fouled in a one-and-one situation is 1.4. Use this information to calculate what Isaac's free-throw shooting percentage is.

Hints and Tips: We'll use the symbols "p" and "E" to represent what we're talking about. Let:

p = success probability whenever Isaac shoots a free throw

E = expected value for number of points scored in a one-and-one situation

In Ann's case, we were given that p = 0.82, and we used that information to draw a tree diagram and solve for E. Conversely, in Isaac's case we know that E = 1.4, and we want to find the value of p. To do this, use exactly the same procedure that you used in Scenario 1, except instead of working with the number 0.82, simply use the symbol p in your tree diagram.

Use this tree to write an equation involving p and E. Plug in the value of E that you were given for Isaac into the equation and solve for p. (Recall from previous math courses that to solve an equation of this nature you'll need to use the quadratic formula. If you've forgotten the formula simply do a Google search for "quadratic equation solver" - there are lots of good, free ones out there for you to use. You have to get the equation in the right format, but once you do that you can use a solver to do all calculations.) You'll find that there are actually two solutions for p. However, in this case p is a probability and we know that probabilities must be values between 0 and 1 so only one of the values is a valid answer.

What is Isaac's probability of success when shooting free throws? (Give your answer correct to three decimal places.)