Independent solution

How to solve this Uniform Distribution question

Answer in brief

The two probability statements identify the uniform support as [4,20]. After conditioning on values above 6, the remaining interval has length 14, and the favorable part from 6 to 10 has length 4. Their ratio is 4/14=2/7, so choice C is correct.

Setup

Setup

Let the unknown uniform support be [a,b] and translate each supplied probability into an interval-length ratio.

12−ab−a=0.50\frac{12-a}{b-a}=0.50
6−ab−a=1−0.875=0.125\frac{6-a}{b-a}=1-0.875=0.125

Model

Model

Divide the first relation by the second to remove the unknown support length and solve for the lower endpoint.

12−a6−a=0.500.125=4\frac{12-a}{6-a}=\frac{0.50}{0.125}=4
12−a=24−4a12-a=24-4a
a=4a=4

Compute

Compute

Use either probability relation to recover the upper endpoint, then form the conditional interval ratio.

12−4b−4=0.50⟹b=20\frac{12-4}{b-4}=0.50\quad\Longrightarrow\quad b=20
Pr⁡(X<10∣X>6)=Pr⁡(6<X<10)Pr⁡(X>6)\Pr(X<10\mid X>6)=\frac{\Pr(6<X<10)}{\Pr(X>6)}
(10−6)/16(20−6)/16=27\frac{(10-6)/16}{(20-6)/16}=\frac{2}{7}

Answer

Answer

The requested conditional probability is 2/7.

27(C)\boxed{\frac{2}{7}\quad\text{(C)}}