Question

Where’s the bus? Sally takes the same bus to work every morning. Let X = the amount of

time (in minutes) that she has to wait for the bus on a randomly selected day. The

probability distribution of X can be modeled by a uniform density curve on the interval

from 0 minutes to8 minutes. Find the probability that Sally has to wait between 2 and 5

minutes for the bus.

Short answer

The probability that Sally has to wait between 2 and 5 minutes for the bus is $37.5\%$.

Step-by-step solution

Step 1. Given information.

The distribution is modeled by a uniform distribution on the interval from 0 minutes to 8 minutes.

$a=0,b=8$

Step 2. To find the density curve.

The density curve of the uniform distribution is reciprocal of the difference of the boundaries

$$\begin{gathered} f\left(x\right)=\frac{1}{b-a}=\frac{1}{8-0} \\ f\left(x\right)=\frac{1}{8} \\ f\left(x\right)=0.125 \end{gathered}$$

Step 3. To find the probability.

The probability, that the time is between 2 and 5 minutes, is then the area underneath the density curve between 2 and 5.

The area of a rectangle is the product of the width and the height.

$$\begin{gathered} P(2<X<5)=P(5-2)\frac{1}{8} \\ =3\times \frac{1}{8} \\ =\frac{3}{8} \\ =0.375 \\ =37.5\% \end{gathered}$$