Question

Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. Let $X=$the time, in minutes, it takes a student to finish a quiz. Then $X~U(6,15)$Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. Then find the probability that a different student needs at least eight minutes to finish the quiz given that she has already taken more than seven minutes.

Short answer

The probability that a randomly selected student needs at least eight minutes to complete the quiz is $0.7778$.

The probability that a different student needs at least eight minutes to finish the quiz given that she has already taken more than seven minutes is $0.875$.

Step-by-step solution

Given Information

Given in the question that $X~U(6,15)$

Where$a=6$and$b=15$

The probability density function 

The probability density function will be,

$f\left(x\right)=\frac{1}{b-a}$

$=\frac{1}{15-6}$

$=\frac{1}{9}$

Calculate the value of P(x≥8)

The probability of student needs at least 8 minutes to complete the quiz can be computed as

$P(x\ge 8)=\text{base}x\text{height}$

$=(15-8)\left(\frac{1}{9}\right)$

$=7\times \frac{1}{9}$

$=0.7778$

Calculate the value of P(x>8∣x>7)

The probability of student needs at least 8 minutes to complete the quiz given 7 minutes that is $P(x>8\mid x>7)$can be calculated as below:

The probability density function or height for this probability will be:

$f\left(x\right)=\frac{1}{b-a}$

$=\frac{1}{15-7}$

$=\frac{1}{8}$

Therefore, the value of $P(x>8\mid x>7)$is,

$P(x>8\mid x>7)=\text{base}\times \text{height}$

$=(15-8)\times \left(\frac{1}{8}\right)$

$=7\times \frac{1}{8}$

$=0.875$