Lesson Solving problems on Binomial distribution with Technology (using MS Excel)
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<H2>Solving problems on Binomial distribution with Technology (using MS Excel)</H2> In this lesson, I teach students on solving Binomial distribution problems with Technology. <H3>Problem 1</H3>If a seed is planted, it has a 65% chance of growing into a healthy plant. If 11 seeds are planted, what is the probability that exactly 4 don't grow? <B>Solution</B> <pre> 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. <U>ANSWER</U> 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 </pre> <H3>Problem 2</H3>Industry standards suggest that 16% of new vehicles require warranty service within the first year. Jones Nissan sold 8 Nissans yesterday. What is the probability that less than three of these vehicles require warranty service within the first year? <B>Solution</B> <pre> This is a binomial distribution type problem where the probability under the question is the sum P = {{{sum (C(8,k)*p^k*q^(8-k),k=0,3)}}} (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 {{{sum(C(8,k)*0.16^k*(1-0.16)^(8-k),k=0,2)}}} = 0.877402. <U>ANSWER</U> 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 </pre> <H3>Problem 3</H3>It is estimated that 20% of luxury cars manufactured in 2019 were silver. A car dealership typically sells 20 luxury cars per month. Find the probability that more than 10 of the luxury cars sold per month are silver. <B>Solution</B> <pre> This is a binomial distribution type problem where the probability under the question is the sum P = {{{sum (C(20,k)*p^k*q^(20-k),k=11,20)}}} (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 - {{{sum(C(20,k)*p^k*q^(20-k),k=0,10)}}}. (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 {{{sum(C(20,k)*0.2^k*(1-0.2)^(20-k),k=0,10)}}} = 0.999437. In this way, the value of (2) is equal to 1 - 0.999437 = 0.000563 (approximately). <U>ANSWER</U> 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 </pre> <H3>Problem 4</H3>Of all the new vehicles of a certain model that are sold, 20% require repairs to be done under warranty during the first year of service. A particular dealership sells 14 such vehicles. <pre> (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? </pre> <B>Solution</B> I will solve part (a) first. <pre> 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) = {{{sum (C(14,k)*p^k*q^(14-k),k=0,4)}}} (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 = {{{sum(C(14,k)*0.2^k*(1-0.2)^(14-k),k=0,4)}}} = 0.87016. <U>ANSWER</U> </pre> Thus part (a) is just solved using Technology. ------------- Now I will solve part (b). <pre> 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) = {{{sum (C(14,k)*p^k*q^(14-k),k=3,14)}}} (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 {{{sum (C(14,k)*p^k*q^(14-k),k=3,14)}}} = 1 - {{{sum (C(14,k)*p^k*q^(14-k),k=0,2)}}} (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 {{{sum (C(14,k)*p^k*q^(14-k),k=0,2)}}} = BINOM.DIST(2, 14, 0.2, TRUE) = 0.448051. In this way, the value of (2) is equal to 1 - 0.448051 = 0.551949. <U>ANSWER</U>. The probability under the question (b) is P = 0.551949. </pre> Thus the problem is solved using Technology. <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 Instead of using Excel function BINOM.DIST, you can use similar function <U>binompdf</U> of your pocket calculator TI-83 or TI-84, with the same success. </pre> <H3>Problem 5</H3>Assume that 17% of people are left-handed. If we select 5 people at random, find the probability of each outcome described below. <pre> 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. </pre> <B>Solution</B> 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> <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 </pre> My other Additional lessons on Probability in this site are - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Sample-space-conception-problems-REVISITED.lesson>Sample space conception problems REVISITED</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Solving-probability-problems-using-complementary-probability-REVISITED.lesson>Solving probability problems using complementary probability REVISITED</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Elementary-Probability-problems-related-to-combinations-REVISITED.lesson>Elementary Probability problems related to combinations REVISITED</A> - 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<A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Miscellaneous-problems-on-Probability.lesson>Miscellaneous problems on Probability</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/The-chances-to-be-rescued-from-an-inhabitant-island.lesson>The chances to be rescued from an inhabitant island</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Analyzing-chains-of-random-events.lesson>Analyzing chains of random events</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Math-expectation-of-winning-in-game-problems.lesson>Math expectation of winning in games problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Math-expectation-of-winning-in-lottery-problems.lesson>Math expectation of winning in lottery problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Math-expectation-of-winning-in-games-with-rolling-pair-of-dice.lesson>Math expectation of winning in games with rolling pair of dice</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Problems-on-uniformly-distributed-random-variables.lesson>Problems on uniformly distributed random variables</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Classic-problem-of-Elementary-Geometric-Probability-theory.lesson>Classic problems of Elementary Geometric Probability theory</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Twisted-probability-problems.lesson>Twisted probability problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/Entertainment-probability-problems.lesson>Entertainment probability problems</A> - <A HREF=https://www.algebra.com/algebra/homework/Probability-and-statistics/OVERVIEW-of-my-Additional-lessons-on-Probability.lesson>OVERVIEW of my additional lessons on Probability</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - 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