   # Finding a Probability In Exercises 31-36, sketch the graph of the probability density function over the indicated interval and find each probability. f ( x ) = 1 50 ( 10 − x ) , [ 0 , 10 ] (a) P ( 0 &lt; x &lt; 2 ) (b) P ( x ≥ 7 ) (c) P ( x ≤ 5 ) (d) P ( 8 &lt; x &lt; 9 ) ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
Publisher: Cengage Learning
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
Publisher: Cengage Learning
ISBN: 9781305860919
Chapter 9, Problem 33RE
Textbook Problem
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## Finding a Probability In Exercises 31-36, sketch the graph of the probability density function over the indicated interval and find each probability. f ( x ) = 1 50 ( 10 − x ) , [ 0 , 10 ] (a) P ( 0 < x < 2 ) (b) P ( x ≥ 7 ) (c) P ( x ≤ 5 ) (d) P ( 8 < x < 9 )

To determine

To graph: The probability density function f(x)=150(10x) over the interval [0,10] and calculate the probabilities (a) P(0<x<2) (b) P(x7), (c) P(x5),(d) P(8<x<9).

### Explanation of Solution

Given Information:

The probability density function f(x)=150(10x) over the interval [0,10].

Graph:

Consider the provided probability density function.

f(x)=150(10x)

Use the ti-83 graphing calculator to construct the graph of the function.

Step 1: Open the ti-83 graphing calculator.

Step 2: Press [Y=] and enter the function Y1=150(10x).

Step 3: Press the [WINDOW] key and adjust the scale.

Xmin=0Xmax=10Xscl=1

And

Ymin=0Ymax=0.5Yscl=0.1

Step 4: Press the [GRAPH] key.

The graph of the function f(x)=150(10x) is obtained as,

Now, calculate the probabilities.

Consider the provided probability density function f(x)=150(10x).

Use the formula P(cxd)=cdf(x) dx to calculate the required probabilities. So,

P(c<x<d)=cd150(10x) dx

Use the formula xndx=xn+1n+1 and integrate.

cd150(10x) dx=150(10xx22)cd

(a)

To calculate P(0<x<2), use the Fundamental theorem abf(x) dx=F(b)F(a) and apply the limits on 150(10xx22)cd.

150(10xx22)02=150(10(2)2220)=150(202)=1850=925

Thus, the probability of P(0<x<2) is 925.

(b)

To calculate P(x7), use the Fundamental theorem abf(x) dx=F(b)F(a) and apply the limits on 150(10xx22)cd.

150(10xx22)710=150((10(10)1022)(10(7)722))=150((10050)(70492))=150(5070+492)=150(20+492)

Simplify

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