1. ## A basketball player attempts a free throw...

A basketball player attempts a free throw. If he is successful, he can attempt a second free throw. If P is his probability of a successful free throw, and if his probability of making 0 points is equal to that of making 2 points, what is P? (Round the answer to the nearest thousandth.)

2. Originally Posted by ceasar_19134
A basketball player attempts a free throw. If he is successful, he can attempt a second free throw. If P is his probability of a successful free throw, and if his probability of making 0 points is equal to that of making 2 points, what is P? (Round the answer to the nearest thousandth.)
Let's say that: $\displaystyle P=\frac{1}{x}$

So the odds of making a shot are: $\displaystyle \frac{1}{x}$

The odds of making two shots are: $\displaystyle \frac{1}{x}\times \frac{1}{x}=\frac{1}{x^2}$

The odds of missing the first shot are: $\displaystyle 1-\frac{1}{x}=\frac{x-1}{x}$

Which means that: $\displaystyle \frac{x-1}{x}=\frac{1}{x^2}$

Multiply both sides by $\displaystyle x$ to get: $\displaystyle x-1=\frac{1}{x}$

Mutiply again: $\displaystyle x^2-x=1$

Subtract 1 from both sides: $\displaystyle x^2-x-1=0$

Use the quadratic equation: $\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Substitute: $\displaystyle x=\frac{-(-1)\pm\sqrt{(-1)^2-4(1)(-1)}}{2(1)}$

Solve: $\displaystyle x=\frac{1\pm\sqrt{5}}{2}$

That gives that: $\displaystyle x=1.61803399, -0.618033989$

Since it can't be negative, we know that: $\displaystyle x=1.61803399$

Remember: $\displaystyle P=\frac{1}{x}$

Substitute: $\displaystyle P=\frac{1}{1.61803399}$

Solve: $\displaystyle P=0.618033988$

Round to get: $\displaystyle \boxed{P\approx 0.618}$

BUT CHECK MY ARITHMETIC