







Liza works for a call center 
20191031 

From Timothy: Liza knows that in the long term, she has a 65% chance of making a sale when calling on customers. One morning, he makes Six calls.
a. What is her probability of making three calls?
b. What is her probability of making sales fewer than three? Answered by Penny Nom. 





A binomial probability question 
20191008 

From venni: The Medassist Pharmaceutical Company receives large shipments of aspirin
tablets and uses this acceptance sampling plan: Randomly select and test 24
tablets, then accept the whole batch if there is only one or none that doesn’t
meet the required specifications. If a particular shipment of thousands of
aspirin tablets actually has a 4% rate of defects, what is the probability that
this whole shipment will be accepted? Answered by Penny Nom. 





The binomial distribution 
20191001 

From PY: An online game called ‘Shop Quiz’ is held by an ecommerce platform, from Monday to Friday
every week. It consists of 8 multiple choice questions (MCQ) and each question has four options (A, B, C, D). Only one option is the correct answer. People who are able to correctly answer all 8
questions are winners and will be awarded a number of online shopping credits.
Let X represent the number of questions that a person can answer correctly in a ‘Shop Quiz’.
1) Explain why Binomial distribution might NOT be a suitable distribution for the random variable
X.
Mr. Saul likes playing the quiz, however, he is afraid that he might not have the necessary knowledge to answer the quiz questions. (The quiz questions cover a variety of topics including science, history, entertainment, sports and geography, etc.) Therefore, he tries to win the game by simply guessing the answers to each question. Answered by Penny Nom. 





(2018^2019)(2019^2018) 
20190123 

From Nazrul: Find the remainder when (2018^2019)(2019^2018) is divided by 4? Please show me the process. Answered by Harley Weston. 





A binomial expansion 
20190105 

From shifali: if the binomial expansion of (mnx)^3 is 1+9x+ find the values of m & n Answered by Penny Nom. 





nC0 + nC1 + nC2 + .... + nCn = 2^n 
20180219 

From bristal: (QQ) Prove, nC0 + nC1 + nC2 + .... + nCn = 2^n. Answered by Penny Nom. 





Expand and simplify [x3][x+3] 
20140716 

From symion: Expand and simplify [x3][x+3] Answered by Penny Nom. 





Identify each polynomial by its degree and number of terms 
20110110 

From betty: Write in standard form and identify each polynomial by its degree and number of terms. How do you do this? Answered by Penny Nom. 





A binomial random variable 
20101129 

From yvette: a binomial random variable has a mean equal to 200 and standard deviation of 10. find the values of n and p. Answered by Penny Nom. 





(x + a)^2 
20100604 

From a: If the 3rd , 4th, 5th terms in the expansion of (x + a)powe "n" are 720, 1080, 810 respectively, then
find the values of x, a and n. Answered by Penny Nom. 





Sample size 
20100329 

From Rae: What sample size was needed to obtain an error range of 2% if the following statement was made? "75% of the workers support the proposed benefit package. These results are considered accurate to within + or  2%, 18 out of 20 times. This seems like a straight forward question but I'm getting it wrong. Could you please help me out even just the set up would be appreciated so I can see if that's where I'm going wrong. Thanks Answered by Harley Weston. 





Binomial probability 
20100319 

From Rana: According to one study, it has been claimed that 65% of all single men in Montreal would welcome a
woman taking the initiative in asking for a date. You decide to challenge this study by performing a little
experiment of your own. You randomly select 15 single men from around Montreal and ask them if they
would be comfortable with a woman asking them out for a date. If the claim made by the study was
true, what is the probability that:
a) Exactly 9 men would say yes?
b) Exactly 6 men would say no?
c) More than 11 men would say yes?
d) At most 4 men would say no? Answered by Penny Nom. 





0.999 ^ (500) 
20100307 

From debra: I just need to know how to solve the following problem without using a calculator: .999 ^ (500). I know the answer is .606, I just want to do it by hand since I can't use a calculator on my test. Answered by Penny Nom and Claude Tardif. 





The binomial distribution 
20100227 

From Jessica: A small mobile phone retailer has found that one of their phones has a 12% probability of being faulty and a replacement having to be provided for the customer. They have just
received a trial order for 10 phones from their biggest customer who will take their business elsewhere if 20% or more items are faulty.
i) what is the probability that they will lose their biggest customer? Answered by Penny Nom. 





The binomial theorem 
20100121 

From Laura: Using the fact that (1 + x)^4 * (1 + x)^9 = (1 + x)^13 show (4C0 * 9C4 + 4C1*9C3 + 4C2*9C2 + 4C3*9C1 + 4C4*9C0) = 13C4 Answered by Harley Weston. 





nC0 + nC1 + nC2 + ... + nCn = 2^n 
20090615 

From Chinonyerem: For n >= 1, derive the identity
nC0 + nC1 + nC2 + ... + nCn = 2^n
[Hint: Let a = b = 1 in the binomial theorem] Answered by Penny Nom. 





A multiple choice exam 
20080913 

From Phalange: A multiple choice exam consists of 12 questions, each having 5 possible answers. To pass, you must answer at least 9 out of 12 questions correctly. What is the probability of passing if:
a. You go into the exam without knowing a thing, and have to resort to pure guessing?
b. You have studied enough so that on each question, 3 choices can be eliminated. But then you have to make a pure guess between the remaining 2 choices.
c. You have studied enough so that you know for sure the correct answer on 2 questions. For the remaining 10 questions you have to resort to pure guessing. Answered by Harley Weston. 





n choose r equals n1 choose r plus n1 choose r  1 
20080714 

From fae: Prove that
( n ) = ( n – 1) + ( n  1 )
( r ) ( r ) (r1)
NOTE: the ( ) should be one for n taken r and so on. but there is no one big ( ) that will cater two lines Answered by Janice Cotcher. 





A coin is tossed 30 times 
20080413 

From seulki: coin tossed 30 times, what is the probability that the heads show up fewer than 17 times? Answered by Harley Weston. 





Probability and Merta trains 
20080327 

From Lorie: 4. Merta claims that 74% of its trains are on time.
a. Find the probability that among the 60 trains, 38 or fewer arrived on time. Answered by Harley Weston. 





Multiple guess 
20080227 

From debbie: What is the probability of getting 6 out of 10 (a,b,c,d,e) questions correct, by random selection? Answered by Stephen La Rocque. 





A binomial probability problem 
20080108 

From rougi: According to statistics of the company about 4% of the people who buy tickets do not arrive to the flight . The company sells 205 tickets to plane that has 200 places . Use the binomial distribution to calculate what is the probability that all the people who come to the flight get a place. Answered by Harley Weston. 





Binomial probability 
20071110 

From areej: In any table of binomial probability with n trials, why is it that the probabilities for k=n are not tabulated??? Answered by Harley Weston. 





A multiple choice exam 
20071021 

From jon: a student is taking a multiple choice exam in which each question has four choices.
assuming that she has no knowledge of the correct answers to any of the questions, she
has decided on a strategy in which she will place four balls (marked A, B, C, D) into a box. she randomly
selects one ball for each question and replaced the ball in the box. the marking will determine her answer to the question.
there are five multiple choice questions on the exam. what's the probability that she will ... Answered by Stephen La Rocque and Harley Weston. 





The normal approximation to the binomial 
20070930 

From m.j.: Slot Machines The probability of winning on a slot machine is 5%. If a person plays the machine 500 times, find the probability of winning 30 times. Use the normal approximation to the binomial distribution. Answered by Harley Weston. 





Standard deviation 
20070901 

From Pat: I have looked over your math question and mines is not there. for a binomiial distribution with parmeter n and p, the mean is np and the standard deviation is np(1p). I understand how to get the np but not understanding how to get the (1p). For example np=(40)(0.48)(10.48)=3.160. How can I get the (10.48) answer please Answered by Harley Weston. 





Find the sample size needed 
20070513 

From Mini: Find the sample size needed to be 98% confident thata marketing survey on the proportion of shoppers who use the internet for holiday shopping is accurate within a margin of error of 0.02. Assume that the conditions for a binomial distribution are met, and that a current estimate for a sample proportion does not exist. Answered by Penny Nom. 





the sum form r = 0 to r = m of C(n+r,n)=? 
20070415 

From Aastha: FIND THE VALUE OF:
the sum form r = 0 to r = m of C(n+r,n)=? Answered by Penny Nom. 





A hypothesis test 
20070409 

From Katrina: I have already tried to do this problem but im having a very had time with
it. Can you please help me.
Glamour Magazine sponsored a survey of 2500 prospective brides and
found that 60% of them spent less than $750 on their wedding gown.
Use a 0.01 significance level to test the claim that less than 62% of brides
spend less than $750 on their wedding gown. How are the results
affected if it is learned that the responses were obtained from magazine
readers who decided to respond to the survey through an Internet Web
site? Answered by Penny Nom. 





Colour blindness 
20070329 

From Katrina: 9% of men and .25% of woman can't distinguish between
the colors red and green. This is the type of color blindness
that causes problems with traffic lights. If 6 men are randomly
selected for a study of traffic signal perceptions, find the probability
that exactly two of them cannot distinguish between red and green. Answered by Stephen La Rocque. 





Acceptance sampling 
20070329 

From Katrina: The Medassist Pharmaceutical Company recieves large shipments of
asprin tablets and uses this acceptance sampling plan: Randomly select
and test 24 tablets, then accept the whole batch if there is only one or
none that doesnt meet the required specifications. If a particular shipment
of thousands of asprin tablets actually has a 4% rate of defects, what is
the probability that this whole shipment will be accepted? Answered by Penny Nom. 





What is the expected number of ripe and ready to eat watermelons 
20061129 

From James: An agricultural cooperative claims 95 percent of the watermelons shipped out are ripe and ready to eat. If 20 watermelons are shipped out , what is the probability that the number of watermelon that are ripe and ready to is (i) exactly 14 (ii) more than 18 (iii) of the 20 watermelons that are shipped what is the expected number of ripe and ready to eat watermelons Answered by Penny Nom. 





Sally plays roulette 
20061121 

From Sherry: Sally plans to bet 100 bets of $1.00 each on red roulette. the probability of the ball landing on red is 18/38 what is the probability that Sally will win at least half the time Answered by Penny Nom. 





Prove that 2nCn is less than 4n, for all positive integers n? 
20061001 

From Anna: How can I prove that 2nCn is less than 4n, for all positive integers n? Answered by Penny Nom. 





Square roots in a binomial expansion 
20060911 

From Sydney: (√x + 5)^{4} expanded using the binomial theorem Answered by Penny Nom. 





What are the 3rd and 4th terms of (2xy)^7? 
20060618 

From April: What are the 3rd and 4th terms of this sequence: (2xy)^{7}?
I'm having an issue with this...is there any easier way to get it without completely factoring the whoooole thing out? Answered by Penny Nom. 





The coefficient of variation 
20060520 

From Glenn: What is the correct formula for coefficient of variation for a binomial distribution? Answered by Penny Nom. 





A manufacturer of cotton pins 
20060320 

From Nirmal: A manufacturer of cotton pins knows that 5% of his products are defective. If he sells cotton pins in boxes of 100 and guarantees that not more than 10 pins will be defective, what is the approximate probability that a box will have the guaranteed quantity? Answered by Penny Nom. 





A binomial distribution exercise 
20060121 

From Belinda: In a survey of 15 manufacturing firms, the number of firms that use LIFO (a lastin firstout accounting procedure for inventory) is a binomial random variable x with n=15 and p=0.2.
a) What is the probability that five or fewer firms will be found to use LIFO? Is it unlikely that more than 10 firms will be found to use LIFO? Comment. Answered by Penny Nom. 





A confidence interval 
20060121 

From Jonathan:
I am attempting to calculate how my confidence interval will widen at the 95% confidence level if my response universe increases from 100 to 150 or to 200.
There is a universe of 54,000. I take a 5% sample for a test universe of 2,700
If my "yes" universe is 100, at the 95% confidence level, what is my +/ range? (i.e +/ 3? +/5?)
Historically, 6.6% of the 2,700 you say "yes". I am trying to determine how the confidence interval would change if the number of "yes" responders increased to 150 or to 200.
Answered by Penny Nom. 





5x^2  27x  18 
20060113 

From Katy: How would you factor 5x^{2}  27x  18? Answered by Penny Nom. 





A sample size estimation 
20051203 

From Ivonne: I have to do a research about the behavior of library users. We are going to apply a survey to a population of 1280 students (Management an Economics students) but of course we have to do it to a sample....I need to know the size of my sample. Answered by Penny Nom. 





Discrimination based on gender? 
20050310 

From A student: After being rejected for employment, Kim learns that the Bellevue office has hired only two women among the last 20 new employees. She also learns that the pool of applicants is very large, with an approximately equal number of qualified men and women. Help her address the charge of gender discrimination by finding he probability of getting two or few women when 20 people are hired, assuming that there is no discrimination based on gender. Does the resulting probability really support such a charge? Answered by Penny Nom. 





Computing confidence intervals 
20041126 

From Christie: I was given a question with N=100, sample proportion is 0.1 compute the 95% confidence interval for P? I have tried this several ways but do not know how to do without means, standard deviations, standard error of the mean? I asked my teacher and she said I have all the info I need. Can you help???? Answered by Penny Nom. 





A binomial squared 
20040919 

From John: Can you explain the steps
for me, that get you from (a  b)^{2} to a^{2} 
2ab + b^{2} Answered by Penny Nom. 





Replacement times for TV sets? 
20040331 

From Barb: Replacement times for TV sets are normally distributed with a mean of 8.2 years and a standard deviation of 1.1 years. Estimate the probability that for 250 randomly selected TV sets, at least 15 of them have replacement times greater than 10.0 years.e Answered by Andrei Volodin and Penny Nom. 





Binomial distribution 
20031217 

From Lesley: my daughter is having difficulty with the following formula
P(X=x) = ( n over x) px (1p) nx
The teacher has given them the formula but not taught them how to apply it or understand it. Answered by Penny Nom. 





Newton's binomial theorem 
20030830 

From William: According to page 126 of Murtha & Willard's "Statistics and Calculus" (PrenticeHall, 1973), Newton's binomial theorem can proved inductively. I suppose that was his method, which I would like to see. Answered by Penny Nom. 





A binomial probability 
20030430 

From Erica: In a biathlon, athletes shot at 20 targets. A particular athlete had a probability of 0.18 of missing a target. Assuming a binomial distribution, what is the probability, to the nearest thousandth, that the athlete hit exactly 16 targets? Answered by Andrei Volodin. 





The effectiveness of a drug 
20030422 

From A student: A certain drug is found to be effective 80% of the time. Find the probability of successful treatment in two out of four cases. Answered by Andrei Volodin. 





Successive coefficients in Pascal's Triangle 
20021227 

From Quincy: There is a formula connecting any (k+1) successive coefficients in the nth row of the Pascal Triangle with a coefficient in the (n+k)th row. Find this formula Answered by Penny Nom and Walter Whiteley. 





Successive coefficients in the nth row of Pascal's Triangle 
20020610 

From Tim: There is a formula connecting any (k+1) successive coefficients in the nth row of Pascal's Triangle with a coefficient in the (n+k)th row. find this formula. Answered by Penny Nom. 





Pascal's Triangle 
20020402 

From Brian: It's about (a+b)^{x}. I remember there a triangle with numbers to remember for a faster solution. Can you please teach me? Answered by Penny Nom. 





Rolling 5 sevens before rolling a six or an eight 
20020120 

From Tony: When rolling 2 dice, what is the probability of rolling 5 sevens before rolling a six or an eight? Answered by Andrei Volodin and Penny Nom. 





Cinderella clothes 
20011211 

From A student: If cinderella clothes, inc. has determined that 0.5% of all incoming phone calls involve complaints, what is the probability that in 200 incoming calls there are more than one complaint? Answered by Andrei Volodin. 





Multinomial theorem 
20011128 

From Murray: Could you please state and explain the multinomial theorem (I already know the binomial theorem etc, to give you an idea of where i am) Answered by Harley Weston. 





(a+b) squared and a squared plus b squared 
20011114 

From Kristen:
(a+b)^{2} = a^{2} + b^{2} Is this true for all whole numbers, and why? Answered by Penny Nom. 





Lucas' theorem 
20011009 

From Tania: How could I demonstrate: nCp is congruent to floor(n/p) (modulo p)? where rCk is a binomial coefficient, rCk = r(r1)...(rk+1)/k(k1)...1, and p is a prime number Answered by Richard McIntosh. 





Binomial probabilities 
20011008 

From Amna: I had a few binomial probability questions which I can not use from the tables as instructed:  If 60 % of television viewers are watching a certain program, what is the probability tha tmore than half of those selected in a random sample of five will be watching the specified program?
 If it is known according to Mendel's Law, that we can expect in teh long run to have 3 white, 1 brown rabbits in every 4 rabbits of a certain type, what is the probability that 2 in a litter of 3 will be white?
 On the average, 2% of the items sold in a department store are returned for refunds. what is the probability that of its next five items sold, at most two will be returned for refunds?
Answered by Leeanne Boehm. 





Multiplying binomials 
20010908 

From A student: (x+1)(x+2)=? (2x1)(x4)=0 please tell me this question!! Answered by Leeanne Boehm. 





(2b+3)(b1) 
20010206 

From Melissa: Alright, lets say you are multiplying (2b+3)(b1) and you are using "FOIL" first you would get : 2b^{2}+2b3b3 then you add liketerms. when multiplying,if the bases are the same, add the exponents...does the same thing apply when you are adding the results of the multiplication even though it's addition? would the answer be 2b^{3}3b3 or would it be 2b^{2}b3? Answered by Penny Nom. 





A dollar, quarter, dime, nickle and penny 
20010107 

From Sarah: Arnold has a dollar coin, one dime, one quarter, one nickel, and a penny. The number of different sums of money that can be formed using three coins is... Answered by Penny Nom. 





Banana yogurt 
20001103 

From James: A grocery store has 100 cartons of banana yogurt in stock.Each carton contains 12 cup of banana yogurts.The probability that a cup has fewer than 20 banana chunks in it is 10 %. So,What is the probability that between 15 and 25 (inclusive) cartons out of the 100 cartons have exactly 3 cups with fewer than 20 banana chunks? Answered by Harley Weston. 





Independent tests 
20001007 

From A student: If the falsepositive rate of each test in a battery of tests is 0.05, how many independent tests can be included in the battery if we want the probability of obtaining at least one falsepositive result to be at most 0.2? Answered by Harley Weston. 





649 
20000914 

From Steve: In our state lottery you must choose 6 numbers (149). How many different combinations are there? They can be in any order. Answered by Harley Weston. 





An expected value 
20000424 

From Carl Pride: Suppose that in a statistics class of size 23, each student has a probability of passing of 73 percent. sample: What is the expected number of students who will pass?? Answered by Harley Weston. 





Binomial coefficients 
20000321 

From Howard Lutz: How do you find each successive numerical term in this equation y+dy=(x+dx)^{5} =x^{5}+5*x^{4}dx+10*x^{3}(dx)^{2}+10*x^^{2}(dx)^{3}+5*x(dx)^{4}+(dx)^{5} I would appreciate an explanation of the method to find the numeric coefficient in a binomial expansion Answered by Penny Nom. 





Multinomial coefficients 
19991203 

From Suraj Das: Is there a formula for the expansion of (a+b+c) to the nth power? Does it have to do with Pascal's triangle? Answered by Penny Nom. 





The Binomial Theorem for rational exponents 
19990415 

From Angela Evans: The full question is this: Isaac Newton generalized the Binomial Theorem to rational exponents. That is, he derived series of expansions for such expressions as (x+y)^{3} (x+y)^{2/3} (x+y)^{5/6} What did Newton find? What are the first four terms of the series expansions of binomials above? How can this extended Binomial Thrm. be used to aid in calculations? Answered by Penny Nom. 





Factoring 
19990330 

From Maggie Stephens: I don't know anything about factoring would you plese help me. 3x^{4}  48 54x^{6} + 16y^{3} 1258x^{3} 12x^{2}  36x + 27 9  81x^{2} a^{3} + b^{3}c^{3} I would greatly appreciate any help you can give me thanks. Answered by Jack LeSage. 





Cannonballs 
19990127 

From Roger King: How many cannonballs can be stacked in a triangular pyramid? Answered by Penny Nom. 





Three keys 
19981126 

From Karen Chan: A man has a bunch of three keys, only one of which fits the lock of his front door. When he comes home in the dark he tries the keys at random until he finds the one fits. Find the probability that in a week of five nights, he tries the right key first on at least one night. Answered by Penny Nom. 





What's the next term? 
19981112 

From Ilia: What the formula and explonation for formula for next patterns: 1) 1, 4, 10, 20, 35, 56, 84,... 2) 0, 6, 30, 90, 210, 420,... Thanks! Answered by Penny Nom. 





What is the variance of the difference of two binomials? 
19960208 

From Chris Johnson: My coworkers and I have come up with different estimates of the zstatistic, and are in particular disagreement over the calculation of Variance for this problem. I am trying to find out, with a five percent level of significance, whether the new form yields a higher rate of return than the old form. Any thoughts, comments, or solutions? Answered by Harley Weston. 





(x+y)^n = x^n + y^n + Z 
20040318 

From JeanSébastien: Bonjour,
je me m’appelle
JeanSébastien et je suis étudiant en secondaire
4. Je me demandais s’il est possible de développer le terme
(x+y)^{n}, lorsque n est un entier positif. Il est évident
que x^{n}+y^{n} fera partie de la réponse mais comment définir le reste (Z
dans l’exemple cidessous)?
(x+y)^{2} = x^{2}+y^{2}+2xy
(x+y)^{n} = x^{n}+y^{n}+ Z
P.S. : Une preuve mathématique démontrant la possibilité ou
l’impossibilité du problème serait grandement appréciée,
merci. Answered by Claude Tardif. 

