Textbook Problem

Probability In Exercises 91 and 92, the function

$f\left(x\right)=k{x}^n{\left(1-x\right)}^m,0\le x\le 1$

Where n>0, m>0, and k is a constant, can be used to represent various probability distributions. If k is chosen such that

${\displaystyle {\int }_0^{1}f\left(x\right)dx=1}$ then the probability that x will fall between a and b $\left(0\le a\le b\le 1\right)$ is

$P_{a,b}={\displaystyle {\int }_a^{b}f\left(x\right)dx}$.

The probability that ore samples taken from a region contain between 100a% and 100b% iron is

$P_{a,b}={\displaystyle {\int }_a^b\frac{1155}{32}{x}^3{\left(1-x\right)}^{3/2}dx}$

Where x represents the proportion of iron. (See figure.)

What is the probability that a sample will contain between 0% and 25% iron?

What is the probability that a sample will contain between 50% and 100% iron?