Let T be the time needed to complete a job at a certain factory. By using the historical data, we know that P(T≤t)={116t2for 0≤t≤41for t≥4 a. Find the probability that the job is completed in less than one hour, i.e., find P(T≤1). b. Find the probability that the job needs more than 2 hours. c. Find the probability that 1≤T≤3.

Let T be the time needed to complete a job at a certain factory.&nbsp;P(T≤t)={116t2for&nbsp;0≤t≤41for&nbsp;t≥4&nbsp;&nbsp;&nbsp;(a) From the given information&nbsp;P(T≤1)=116(1)2=116&nbsp;P(T≤1)=0.0625&nbsp;(b) Let&nbsp;P(T≥2)=1−P(T≤2)&nbsp;=1−116(2)2=1−116(4)=1−14=34&nbsp;P(T≥2)=0.75&nbsp;(c)&nbsp;P(1≤T≤3)=P(T≤3)−P(T≤1)&nbsp;=116(3)2−116(1)2=116(9−1)=116(8)=12&nbsp;P(1≤T≤3)=0.5

Explained: "Let T be the time needed to complete..."

nemired9

Answered question

2021-12-11

Let T be the time needed to complete a job at a certain factory. By using the historical data, we know that

P (T \leq t) = {\begin{cases} \frac{1}{16} t^{2} & f o r 0 \leq t \leq 4 \\ 1 & f o r t \geq 4 \end{cases}

a. Find the probability that the job is completed in less than one hour, i.e., find

P (T \leq 1)

.
b. Find the probability that the job needs more than 2 hours.
c. Find the probability that

1 \leq T \leq 3

Answer & Explanation

Lynne Trussell

Beginner2021-12-12Added 32 answers

Let T be the time needed to complete a job at a certain factory.
$P (T \leq t) = {\begin{cases} \frac{1}{16} t^{2} & f o r 0 \leq t \leq 4 \\ 1 & f o r t \geq 4 \end{cases}$

(a) From the given information
$P (T \leq 1) = \frac{1}{16} {(1)}^{2} = \frac{1}{16}$
$P (T \leq 1) = 0.0625$

(b) Let $P (T \geq 2) = 1 - P (T \leq 2)$ $= 1 - \frac{1}{16} {(2)}^{2}$ $= 1 - \frac{1}{16} (4)$ $= 1 - \frac{1}{4}$ $= \frac{3}{4}$
$P (T \geq 2) = 0.75$

(c) $P (1 \leq T \leq 3) = P (T \leq 3) - P (T \leq 1)$ $= \frac{1}{16} {(3)}^{2} - \frac{1}{16} {(1)}^{2}$ $= \frac{1}{16} (9 - 1)$ $= \frac{1}{16} (8)$ $= \frac{1}{2}$
$P (1 \leq T \leq 3) = 0.5$

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