Question

a statistics professor plans classes so carefully that the lengths of her classes are uniformly distributed between 46.0 and 56.0 minutes. find the probability that a given class period runs between 51.25 and 51.5 minutes.
find the probability of selecting a class that runs between 51.25 and 51.5 minutes
(round to three decimal places as needed.)

Explanation:

Step1: Identify uniform - distribution formula

For a uniform distribution on the interval $[a,b]$, the probability density function is $f(x)=\frac{1}{b - a}$ for $a\leq x\leq b$. Here, $a = 46.0$ and $b = 56.0$, so $f(x)=\frac{1}{56.0 - 46.0}=\frac{1}{10}$.

Step2: Calculate the probability

The probability $P(c\leq X\leq d)$ for a uniform distribution is given by $P(c\leq X\leq d)=\frac{d - c}{b - a}$. Here, $c = 51.25$, $d = 51.5$, $a = 46.0$, and $b = 56.0$. So $P(51.25\leq X\leq 51.5)=\frac{51.5 - 51.25}{56.0 - 46.0}$.
$P(51.25\leq X\leq 51.5)=\frac{0.25}{10}=0.025$.

Answer:

$0.025$