Question 1166570
.
According to Masterfoods, the company that manufactures M&M’s, 12% of peanut M&M’s are brown, 15% are yellow, 
12% are red, 23% are blue, 23% are orange and 15% are green. You randomly select six peanut M&M’s 
from an extra-large bag of the candies. (Round all probabilities below to four decimal places; 
i.e. your answer should look like 0.1234, not 0.1234444 or 12.34%.)

(a) Compute the probability that exactly three of the six M&M’s are blue.
(b) Compute the probability that three or four of the six M&M’s are blue.
(c) Compute the probability that at most three of the six M&M’s are blue.
(d) Compute the probability that at least three of the six M&M’s are blue.
(e) If you repeatedly select random samples of six peanut M&M’s, on average how many do you expect to be blue? 
    (Round your answer to two decimal places.)
(f) With what standard deviation? (Round your answer to two decimal places.)
~~~~~~~~~~~~~~~~~~~~~~~~~


<pre>
(a)  In this case, we have a binomial distribution with n=6 trials, k=3 successful trials.
     p=0.23 probability of the individual success in each trial.  Apply the standard formula for this probability

        P = {{{C[6]^3*0.23^3*(1-0.23)^3}}} = {{{((6*5*4)/(1*2*3))*0.23^3*0.77^3}}} = 0.1111 (rounded).   <U>ANSWER</U>



(b)  In this case

        P = P(3) + P(4) = {{{C[6]^3*0.23^3*0.77^3}}} + {{{C[6]^4*0.23^4*0.77^2}}} = 

                        = {{{20*0.23^3*0.77^3 + 15*0.23^4*0.77^2}}} = 0.1360  (rounded).    <U>ANSWER</U>



(c)  Continue in the same style

        P = P(0) + P(1) + P(2) + P(3) = . . . = 0.972  (rounded).    <U>ANSWER</U>


     To replace monotonic calculations, you may use very convenient online calculator
     https://stattrek.com/online-calculator/binomial/



(d)  Continue in the same style

        P = P(3) + P(4) + P(5) + P(6) = 1 - ( P(0) + P(1) + P(2) ) = 1 - 0.8609 = 0.1391  (rounded).    <U>ANSWER</U>


     To replace monotonic calculations, you may use very convenient online calculator
     https://stattrek.com/online-calculator/binomial/



(e)  Mathematical expectation is  n*p = 6*0.23 = 1.38 blue peanuts.



(f)  With the standard deviation  SD = {{{sqrt(n*p*(1-p))}}} = {{{sqrt(6*0.23*(1-0.23))}}} = 1.0308  (rounded).    <U>ANSWER</U>
</pre>

Solved: all questions are answered.