Question 1183439
.
A city has four fire engines. For each fire engine the probability that it is available is 0.9. 
A fire accident happens somewhere in the city one Thursday afternoon. 
It is determined that fire engines will be needed:
(i) What is the chance that exactly 2 engines are available?
(ii) What is the probability that at least two of the engines are available?
(iii) What is the chance that none of the engines are available?
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All three problems are the binomial distribution type of problems.



<pre>
(i)   The number of trials n= 4; the probability of the success trial is 0.9,  and the number of success trials k = 2. 


          P = {{{C[4]^2*0.9^2*(1-0.9)^(4-2)}}} = {{{((4*3)/2)*0.9^2*0.1^2}}} = 0.0486  (rounded).     <U>ANSWER</U>



(ii)  The number of trials n= 4; the probability of the success trial is 0.9,  and the number of success trials k >= 2.


      We need calculate  P(n=4; k>=2; p=0.9).      


      To facilitate calculations, I use an appropriate online (free of charge) calculator at this web-site 

      https://stattrek.com/online-calculator/binomial.aspx


      It provides nice instructions  and  a convenient input and output for all relevant options/cases.


          P(n=4; k>=2; p=0.9) = 0.9963   (rounded).       <U>ANSWER</U>



(iii)  This probability is


          P = {{{(1-0.9)^4}}} = {{{0.1^4}}} = 0.0001.      <U>ANSWER</U>
</pre>

Solved.  &nbsp;&nbsp;&nbsp;&nbsp;// &nbsp;&nbsp;&nbsp;&nbsp;All questions are answered and explained.


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If you want to see many similar &nbsp;(or different) &nbsp;solved problems, &nbsp;look into the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Solving-problems-on-Binomial-distribution-manually.lesson>Simple and simplest probability problems on Binomial distribution</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Typical-binomial-distribution-probability-problems.lesson>Typical binomial distribution probability problems</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/How-to-calculate-binomial-probabilities-using-Technology.lesson>How to calculate Binomial probabilities with Technology (using MS Excel)</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Solving-problems-on-Binomial-distribution-using-Technology.lesson>Solving problems on Binomial distribution with Technology (using MS Excel)</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Solving-problems-on-Binom-distr-with-Technology-%28using-online-solver%29.lesson>Solving problems on Binomial distribution with Technology (using online solver)</A> 

in this site.


After reading these lessons, &nbsp;you will be able to solve such problems on your own, 

which is your &nbsp;PRIMARY &nbsp;MAJOR &nbsp;GOAL &nbsp;visiting this forum &nbsp;(I believe).