Independent solution

How to solve this Normal Distribution question

Setup

Setup

Move all terms to one side and find the two roots of the resulting quadratic.

8X−X2<1⟺X2−8X+1>08X-X^2<1\Longleftrightarrow X^2-8X+1>0
x±=4±15x_{\pm}=4\pm\sqrt{15}

Model

Model

Because the quadratic opens upward, the desired event lies outside the interval between the roots.

{8X−X2<1}={X<4−15}∪{X>4+15}\{8X-X^2<1\}=\{X<4-\sqrt{15}\}\cup\{X>4+\sqrt{15}\}

Compute

Compute

Standardize both boundaries and add the disjoint normal tails.

z−=4−15−52=−2.436491673…z_- =\frac{4-\sqrt{15}-5}{2}=-2.436491673\ldots
z+=4+15−52=1.436491673…z_+ =\frac{4+\sqrt{15}-5}{2}=1.436491673\ldots
Pr⁡=Φ(z−)+1−Φ(z+)=0.0828464975…\Pr=\Phi(z_-)+1-\Phi(z_+)=0.0828464975\ldots

Answer

Answer

The two-tail probability corresponds to the listed value 0.082.

0.082(C)\boxed{0.082\quad\text{(C)}}