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Statistics / Statistics for Engineers and Scientists 4 / Chapter 2.4 / Problem 12E

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

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Three components are randomly sampled, one at a time, from

Chapter 2, Problem 12E

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QUESTION:

Three components are randomly sampled, one at a time, from a large lot. As each component is selected, it is tested. If it passes the test, a success (S) occurs; if it fails the test, a failure (F) occurs. Assume that $80\%$ of the components in the lot will succeed in passing the test. Let $X$ represent the number of successes among the three sampled components.

a. What are the possible values for $X?$

b. Find \(P(X=3)\).

c. The event that the first component fails and the next two succeed is denoted by FSS. Find $P$ (FSS).

d. Find $P(SFS)$ and $P(SSF)$ .

e. Use the results of parts (c) and (d) to find \(P(X=2)\).

f. Find \(P(X=1)\).

g. Find \(P(X=0)\).

h. Find \(\mu_{X}\).

i. Find \(\sigma_{X}^{2}\).

j. Let $Y$ represent the number of successes if four components are sampled. Find \(P(Y=3)\).

Equation Transcription:

$P(X=3)$

$P(X=2)$

$P(X=1)$

$P(X=0)$

${\mu{}}_{X}$

${{\sigma{}}_{X}}^{2}$

$P(Y=3)$

Text Transcription:

P(X=3)

P(X=2)

P(X=1)

P(X=0)

mu_X

sigma_X^2

P(Y=3)

Questions & Answers

QUESTION:

Three components are randomly sampled, one at a time, from a large lot. As each component is selected, it is tested. If it passes the test, a success (S) occurs; if it fails the test, a failure (F) occurs. Assume that $80\%$ of the components in the lot will succeed in passing the test. Let $X$ represent the number of successes among the three sampled components.

a. What are the possible values for $X?$

b. Find \(P(X=3)\).

c. The event that the first component fails and the next two succeed is denoted by FSS. Find $P$ (FSS).

d. Find $P(SFS)$ and $P(SSF)$ .

e. Use the results of parts (c) and (d) to find \(P(X=2)\).

f. Find \(P(X=1)\).

g. Find \(P(X=0)\).

h. Find \(\mu_{X}\).

i. Find \(\sigma_{X}^{2}\).

j. Let $Y$ represent the number of successes if four components are sampled. Find \(P(Y=3)\).

Equation Transcription:

$P(X=3)$

$P(X=2)$

$P(X=1)$

$P(X=0)$

${\mu{}}_{X}$

${{\sigma{}}_{X}}^{2}$

$P(Y=3)$

Text Transcription:

P(X=3)

P(X=2)

P(X=1)

P(X=0)

mu_X

sigma_X^2

P(Y=3)

ANSWER:

Solution

Step 1 of 10

Here we need to find the binomial distribution probabilities

Let x is the number of success

Here n=3, p=0.8, q=0.2

a) hereX $\sim{}$ B(n,p)

The pmf of binomial distribution is P(x)= ${(}_{x}^{n})\ p{\ }^{x}q{\ }^{n-x}$ ; X=0,1,2,3

The possible values of x is 0,1,2,3

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