Question 1202081
.
The probability that a pen drawn at random from a box of pens is defective is 0.1. 
If a sample of 6 pens is taken, find the probability that it will contain:
(a) no defective pens
(b) less than three defective pens.
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        It is a binomial distribution probability problem.

        In this problem,  it is  (implicitly)  assumed that the number of pens in the box is  VERY  LARGE.



<pre>
(a)  P(no defective pens) = P(all 6 pen are no defective) = {{{(1-0.1)^6}}} = {{{0.9^6}}} = 0.531441 (rounded).  <U>ANSWER</U>



(b)  P(less than three defective pens) = P(0 defective pens) + P(1 defective pen) + P(2 defective pens).

    
     P(0 defective pens) = 0.531441   (just found in (a) );


     P(1 defective pen) = {{{C[6]^1*0.1*0.9^5}}} = {{{10*0.1*0.9^5}}} = 0.354294  (rounded);


     P(2 defective pen) = {{{C[6]^2*0.1^2*0.9^4}}} = {{{((6*5)/(1*2))*0.1^2*0.9^4}}} = 0.098415  (rounded).


     The final probability is the sum of the found partial probabilities

         P = 0.531441 + 0.354294 + 0.098415 = 0.98415.    <U>ANSWER</U>
</pre>

Solved.


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If you want to see many other similar &nbsp;(and different) &nbsp;solved problems of this type, &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 from these lessons, &nbsp;you will be able to solve such problems on your own, &nbsp;which is your 

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