Question 1195864
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                        Part  (a)


<pre>
This problem is a binomial distribution with the parameters

    n = 4    (the number of trials);

    k >= 2   (the number of success);

    p = 0.2  (the probability of individual success).


The probability is  

    P = P(2) + P(3) + P(4) = P(n=4, k>=2, p=0.2) = {{{sum(C[4]^k*p^k*(1-p)^(n-k), k=2,4)}}} = {{{sum(C[4]^k*0.2^k*0.8^(4-k), k=2,4)}}}.


     To facilitate my calculations, I used online calculator at this site  https://stattrek.com/online-calculator/binomial.aspx

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


          The resulting number is P = 0.1808  (rounded).    <U>ANSWER</U>
</pre>

Part &nbsp;(a) &nbsp;is complete.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Part &nbsp;(b)


<pre>
The binomial distribution has the following properties:


    The mean of the distribution (μx) is equal to n * P  = 4*0.2 = 0.8.


    The variance (σ2x) is n * P * ( 1 - P ) = 4*0.2*0.8 = 0.64.


    The standard deviation (σx) is sqrt[ n * P * ( 1 - P ) ] = {{{sqrt(0.64)}}} = 0.8.



For the reference, &nbsp;see this web-site


https://stattrek.com/probability-distributions/binomial#:~:text=The%20binomial%20distribution%20has%20the,(%201%20%2D%20P%20)%20%5D.


or your textbook.
</pre>

Part &nbsp;(b) &nbsp;is complete.