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Statistics / Applied Statistics and Probability for Engineers 6 / Chapter 3.6 / Problem 114E

Applied Statistics and Probability for Engineers | 6th Edition | ISBN: 9781118539712 | Authors: Douglas C. Montgomery, George C. Runger

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Consider the endothermic reactions in Exercise 3-32. A

Chapter 3, Problem 114E

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QUESTION:

Consider the endothermic reactions in Exercise 3-32. A total of 20 independent reactions are to be conducted.

(a) What is the probability that exactly 12 reactions result in a final temperature less than 272 K?

(b) What is the probability that at least 19 reactions result in a final temperature less than 272 K?

(c) What is the probability that at least 18 reactions result in a final temperature less than 272 K?

(d) What is the expected number of reactions that result in a final temperature of less than 272 K?

Questions & Answers

QUESTION:

Consider the endothermic reactions in Exercise 3-32. A total of 20 independent reactions are to be conducted.

(a) What is the probability that exactly 12 reactions result in a final temperature less than 272 K?

(b) What is the probability that at least 19 reactions result in a final temperature less than 272 K?

(c) What is the probability that at least 18 reactions result in a final temperature less than 272 K?

(d) What is the expected number of reactions that result in a final temperature of less than 272 K?

ANSWER:

Answer

Step 1 of 4

(a)

Consider the endothermic reactions in given below the table.

A total of $20$ independent reactions is to be conducted.

Final Temperature Conditions	Number of Reactions
266 K	48
271 K	60
274 K	92

We are asked to find the probability that exactly $12$ reactions result in a final temperature less than $272K$ .

Hence the probability mass function PMF is according to the definition of PMF present in the table given below.

Final Temperature Conditions $(X)$	Number of Reactions	PMF $p(x)\ =P(X\ =x)$
266 K	48	$P(X\ =266)\ =\frac{48}{200}\ =0.24$
271 K	60	$P(X\ =271)\ =\frac{60}{200}\ =0.3$
274 K	92	$P(X\ =274)\ =\frac{92}{200}\ =0.46$

Let $X$ denote the number of the reactions result in a final temperature less than $272K$ .

We need to find $P(X\ =12).$

Hence $X$ follows a binomial distribution which we can write,

The random variable $X$ that equals the number of trials that result in a success is a binomial random variable with parameters $0<p<1\ and\ n\ =1,\ 2,\ 3,......$

The probability mass function (PMF) of $X$ is,

$f(x)\ =P(X=x)\ =\left({{\ }_{x}^{n}}\right)\times{}{p}^{x}\times{}(1-p{)}^{n-x}\ \ \ \ \ \ \ \ \ x\ =0,\ 1,\ 2,\ 3,....,n$

………………(1)

The probability $p$ that the reactions result in a final temperature less than $272K$ is from the table,

$p(final\ temperature<\ 272K)\ =p(x=266)+p(x\ =271)$

$p(change<\ 4\ days)\ =0.24+0.3\ =0.54$

We have given $n\ =20.$

Hence the substitute the above values into equation (1),

$f(12)\ =P(X=12)\ =\left({{\ }_{12}^{20}}\right)\times{}{p}^{12}\times{}(1-p{)}^{20-12}\ \$

$P(X=12)\ =\left({{\ }_{12}^{20}}\right)\times{}{0.54}^{12}\times{}(1-0.54{)}^{20-12}\ =0.155\ \$

Hence the probability that exactly $12$ reactions result in a final temperature less than $272K$ is $0.155.$

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