Lesson Binomial distribution problem on M&M peanuts

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Binomial distribution problem on M&M peanuts


Problem 1

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.
    (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?
    (f)   With what standard deviation?

Solution 

(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%5B6%5D%5E3%2A0.23%5E3%2A%281-0.23%29%5E3 = %28%286%2A5%2A4%29%2F%281%2A2%2A3%29%29%2A0.23%5E3%2A0.77%5E3 = 0.1111 (rounded).   ANSWER



(b)  In this case

        P = P(3) + P(4) = C%5B6%5D%5E3%2A0.23%5E3%2A0.77%5E3 + C%5B6%5D%5E4%2A0.23%5E4%2A0.77%5E2 = 

                        = 20%2A0.23%5E3%2A0.77%5E3+%2B+15%2A0.23%5E4%2A0.77%5E2 = 0.1360  (rounded).    ANSWER



(c)  Continue in the same style

        P = P(0) + P(1) + P(2) + P(3) = . . . = 0.972  (rounded).    ANSWER


     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).    ANSWER


     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%28n%2Ap%2A%281-p%29%29 = sqrt%286%2A0.23%2A%281-0.23%29%29 = 1.0308  (rounded).    ANSWER


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