Question 1152881
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            It is a binomial distribution type problem.


            I will use a Technology to solve it quickly.


            I will use the Excel standard function BINOM.DIST to facilitate calculations.

            It has the analogue --- the standard function binompdf in pocket calculators TI-83 and TI-84.



<pre>
(a)  P = {{{sum(C(5,k)*0.2^k*(1-0.2)^(5-k),k=1,5)}}} = 1 - {{{C(5,0)*0.2^0*(1-0.2)^5}}} = 1 - BINOM.DIST(0,5,0.2,FALSE) = 0.67232.



(b)  P = {{{C(5,3)*0.2^3*(1-0.2)^2}}} = BINOM.DIST(3,5,0.2,FALSE) = 0.0512.



(c)  P = {{{sum(C(5,k)*0.2^k*(1-0.2)^(5-k),k=4,5)}}} = 1 - {{{sum(C(5,0)*0.2^k*(1-0.2)^(5-k))}}} = 1 - BINOM.DIST(3,5,0.2,TRUE) = 0.00672.



(d)  P = {{{sum(C(5,k)*0.2^k*(1-0.2)^(5-k),k=0,2)}}} = BINOM.DIST(2,5,0.2,TRUE) = 0.94208.
</pre>

On Excel function BINOM.DIST, see its description everywhere, for example

https://support.office.com/en-us/article/binom-dist-function-c5ae37b6-f39c-4be2-94c2-509a1480770c



On binompdf function for pocket calculators TI-83 and TI-84 see the link

http://users.rowan.edu/~schultzl/ti/binomial.pdf


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If you want to learn in more details on how to use this Technology, look into the lessons

&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 using Technology</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</A> 

in this site.


Look also into this links to the recently solved problem


<A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1152847.html>https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1152847.html</A>


https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1152847.html