# Let T be the time needed to complete a job at a certain factory. By us

Let T be the time needed to complete a job at a certain factory. By using the historical data, we know that
$P(T \le t)=\begin{cases}\frac{1}{16}t^{2}& for\ 0 \le t \le 4\\1& for\ t \geq 4 \end{cases}$
a. Find the probability that the job is completed in less than one hour, i.e., find $$\displaystyle{P}{\left({T}\le{1}\right)}$$.
b. Find the probability that the job needs more than 2 hours.
c. Find the probability that $$\displaystyle{1}\le{T}\le{3}$$.

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Lynne Trussell
Step 1
Given:
Let T be the time needed to complete a job at a certain factory.
We know that,
$P(T \le t)=\begin{cases}\frac{1}{16}t^{2}& for\ 0 \le t \le 4\\1& for\ t \geq 4 \end{cases}$
We want to find,
(a) $$\displaystyle{P}{\left({T}\le{1}\right)}=?$$
(b) $$\displaystyle{P}{\left({T}\geq{2}\right)}=?$$
(c) $$\displaystyle{P}{\left({1}\le{T}\le{3}\right)}=?$$
Step 2
Solution:
(a) From the given information
$$\displaystyle{P}{\left({T}\le{1}\right)}={\frac{{{1}}}{{{16}}}}{\left({1}\right)}^{{{2}}}={\frac{{{1}}}{{{16}}}}$$
$$\displaystyle{P}{\left({T}\le{1}\right)}={0.0625}$$
(b) Let $$\displaystyle{P}{\left({T}\geq{2}\right)}={1}-{P}{\left({T}\le{2}\right)}$$
$$\displaystyle={1}-{\frac{{{1}}}{{{16}}}}{\left({2}\right)}^{{{2}}}$$
$$\displaystyle={1}-{\frac{{{1}}}{{{16}}}}{\left({4}\right)}$$
$$\displaystyle={1}-{\frac{{{1}}}{{{4}}}}$$
$$\displaystyle={\frac{{{3}}}{{{4}}}}$$
$$\displaystyle{P}{\left({T}\geq{2}\right)}={0.75}$$
(c) $$\displaystyle{P}{\left({1}\le{T}\le{3}\right)}={P}{\left({T}\le{3}\right)}-{P}{\left({T}\le{1}\right)}$$
$$\displaystyle={\frac{{{1}}}{{{16}}}}{\left({3}\right)}^{{{2}}}-{\frac{{{1}}}{{{16}}}}{\left({1}\right)}^{{{2}}}$$
$$\displaystyle={\frac{{{1}}}{{{16}}}}{\left({9}-{1}\right)}$$
$$\displaystyle={\frac{{{1}}}{{{16}}}}{\left({8}\right)}$$
$$\displaystyle={\frac{{{1}}}{{{2}}}}$$
$$\displaystyle{P}{\left({1}\le{T}\le{3}\right)}={0.5}$$