Lesson Solving problems on Binomial distribution with Technology (using MS Excel)

Solving problems on Binomial distribution with Technology (using MS Excel)

Problem 1

This is a binomial distribution type problem where p(x) = C(n,x)* p^x * q^(n-x)


    n is equal to 11                    // number of trials
    x is equal to  4                    // number of success trials
    p is the probability of don't grow
    q = 1 - p
    C(n,x) = n! / (x! * (n-x)!)         // binomial coefficient


Notice that the "success" in this case is getting "don't grow" with the probability p = 1 - 0.65 = 0.35.


Thus in your case  P(4, 11, 0.35) = C(11,4) * 0.35^4 * 0.65^7.


Use Excel function BINOM.DIST(4, 11, 0.35, FALSE) = 0.243.    ANSWER


    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

Problem 2

This is a binomial distribution type problem where the probability under the question is the sum


     P =       (1)


The number of trials is              8;
The indexes of success trials        k = 0, 1, 2.
The probability of success trial     p = 0.16;
                                     q = 1 - p
C(n,k) = n! / (k! * (n-k)!)          are binomial coefficients.


The sum  (1)  is a cumulative sum.   


Instead of calculating every term of (1) manually and then summing them up, you may use Excel function 

BINOM.DIST(2, 8, 0.16, TRUE)  to calculate the cumulative sum value


     = 0.877402.      ANSWER


    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

Problem 3

This is a binomial distribution type problem where the probability under the question is the sum


     P =       (1)


The number of trials is              20;
The indexes of success trials        k = 11,12,13,14,15,16,17,18,19,20
The probability of success trial     p = 0.2;
                                     q = 1 - p
C(n,k) = n! / (k! * (n-k)!)          are binomial coefficients.


The sum  (1)  is equal to  1 - .     (2)



Instead of calculating every term of (2) manually and then summing them up, you may use Excel function 

BINOM.DIST(10, 20, 0.2, TRUE)  to calculate the cumulative sum value


     = 0.999437.    


In this way, the value of  (2)  is equal to  1 - 0.999437 = 0.000563 (approximately).    ANSWER


    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

Problem 4

    (a) What is the probability that fewer than five of the 14 vehicles require warranty repairs?

    (b) What is the probability that more than 2 of the 14 vehicles require warranty repairs?

This is a binomial distribution type problem, where the probability under the question is the sum


     P = P(fewer than 5 of 14) = P(0) + P(1) + P(2) + P(3) + P(4) =       (1)


The number of trials is              14;
The indexes of success trials        k = 0,1,2,3,4
The probability of success trial     p = 0.2;
                                     q = 1 - p
C(n,k) = n! / (k! * (n-k)!)          are binomial coefficients.


The sum  (1)  is a cumulative sum.


Instead of calculating every term of (1) individually and then summing them up, you may use Excel function 

BINOM.DIST(4, 14, 0.2, TRUE)  to calculate the cumulative sum value in one click.


In this way,  you get


    P =  = 0.87016.      ANSWER

This is a binomial distribution type problem, where the probability under the question is the sum


     P(more than 2 of 14) = P(3) + P(4) + P(5) + . . . + P(14) =       (1)


The number of trials is              14;
The indexes of success trials        k = 3,4,5, . . . ,14
The probability of success trial     p = 0.2;
                                     q = 1 - p
C(n,k) = n! / (k! * (n-k)!)          are binomial coefficients.



Instead of calculating every term of (1) individually and then summing them up, you may use Excel standard function BINOM.DIST

which allows calculating the entire aggregate of such addends in one click.


But function BINOM.DIST works for so called cumulative sums ONLY, that are the sums of addends for k from 0 to some integer number.


Therefore, I should convert the sum (1) to the cumulative form.


The conversion is this formula  

     = 1 -        (2)


In the right side of (2),  the sum is in the cumulative form, so I can apply the standard Excel function BINOM.DIST to calculate it


     = BINOM.DIST(2, 14, 0.2, TRUE) = 0.448051.


In this way, the value of  (2)  is equal to  1 - 0.448051 = 0.551949.    


ANSWER.  The probability under the question (b) is  P = 0.551949.

    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


    Instead of using Excel function BINOM.DIST, you can use similar function binompdf 
    of your pocket calculator TI-83 or TI-84, with the same success.

Problem 5

    a. There are some lefties (≥1) among the 5 people. 

    b. There are exactly 3 lefties in the group. 

    c. There are at least 4 lefties in the group. 

    d. There are no more than 2 lefties in the group. 

(a)  P =  = 1 -  = 1 - BINOM.DIST(0,5,0.2,FALSE) = 0.67232.



(b)  P =  = BINOM.DIST(3,5,0.2,FALSE) = 0.0512.



(c)  P =  = 1 -  = 1 - BINOM.DIST(3,5,0.2,TRUE) = 0.00672.



(d)  P =  = BINOM.DIST(2,5,0.2,TRUE) = 0.94208.

    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