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1 Steps to Using the Normal Approximation . Here’s an example: suppose you flip a fair coin 100 times and you let X equal the number of heads. Exercise 2. | 2 dhyper dbinom difference 0 2.446417e-16 8.881784e-16 -6.435368e-16 The plot below shows this hypergeometric distribution (blue bars) and its binomial approximation (red). $P(k \; \text{successes in n trials}) = {n\choose k} p^k (1-p)^{n-k}$ The properties of a binomial experiment are: 1) The number of trials $$n$$ is constant. 2 x | {\displaystyle b} Salt and pepper What does it mean? {\displaystyle b} Find the probability of getting 2 heads and 1 tail.Solution to Example 1When we toss a coin we can either get a head $$H$$ or a tail $$T$$.We use the tree diagram including the three tosses to determine the eval(ez_write_tag([[728,90],'analyzemath_com-box-4','ezslot_2',260,'0','0']));sample space $$S$$ of the experiment which is given by: {\displaystyle |x|} A multiple choice test has 20 questions. eval(ez_write_tag([[468,60],'analyzemath_com-banner-1','ezslot_3',367,'0','0']));Example 2A fair coin is tossed 5 times.What is the probability that exactly 3 heads are obtained?Solution to Example 2The coin is tossed 5 times, hence the number of trials is $$n = 5$$.The coin being a fair one, the outcome of a head in one toss has a probability $$p = 0.5$$ and an outcome of a tail in one toss has a probability $$1 - p = 0.5$$The probability of having 3 heads in 5 trials is given by the formula for binomial probabilities above with $$n = 5$$, $$k = 3$$ and $$p = 0.5$$$$\displaystyle P(3 \; \text{heads in 5 trials}) = {5\choose 3} (0.5)^3 (1-0.5)^{5-3} \\ = \displaystyle {5\choose 3} (0.5)^3 (0.5)^{2}$$Use formula for combinations to calculate$$\displaystyle {5\choose 3} = \dfrac{5!}{3!(5-3)!} For example, suppose that we guessed on each of the 100 questions of a multiple-choice test, where each question had one correct answer out of four choices. For example, if Now on to the binomial. Normal Approximation to the Binomial Some variables are continuous—there is no limit to the number of times you could divide their intervals into still smaller ones, although you may round them off for convenience. Part (a): Edexcel Statistics S2 June 2011 Q6a : ExamSolutions - youtube Video. | | Binomial Distribution Overview. We said that our experiment consisted of flipping that coin once. A sample of 800 individuals is selected at random. Generally, the usual rule of thumb is and . | Successes in Pop., M = 500 No. Let’s take some real-life instances where you can use the binomial distribution. How to answer questions on Binomial Expansion? The binomial distribution is a two-parameter family of curves. The binomial distribution is used to model the total number of successes in a fixed number of independent trials that have the same probability of success, such as modeling the probability of a given number of heads in ten flips of a fair coin. α x In these examples the binomial approximations are very good. For the sampling distribution of the sample mean, we learned how to apply the Central Limit Theorem when the underlying distribution is not normal. Conditions for using the formula. Since this is a binomial problem, these are the same things which were identified when working a binomial problem. The mean of the normal approximation to the binomial is . μ = nπ . Name: Example June 10, 2011 The normal distribution can be used to approximate the binomial. In a binomial experiment, you have a number \( n$$ of independent trials and each trial has two possible outcomes or several outcomes that may be reduced to two outcomes.The properties of a binomial experiment are:1) The number of trials $$n$$ is constant.2) Each trial has 2 outcomes (or that can be reduced to 2 outcomes) only: "success" or "failure" , "true" or "false", "head" or "tail", ...3) The probability $$p$$ of a success in each trial must be constant.4) The outcomes of the trials must be independent of each other.Examples of binomial experiments1) Toss a coin $$n = 10$$ times and get $$k = 6$$ heads (success) and $$n - k$$ tails (failure).2) Roll a die $$n = 5$$ times and get $$3$$ "6" (success) and $$n - k$$ "no 6" (failure).3) Out of $$n = 10$$ tools, where each tool has a probability $$p$$ of being "in good working order" (success), select 6 at random and get 4 "in good working order" and 2 "not in working order" (failure).4) A newly developed drug has probability $$p$$ of being effective.Select $$n$$ people who took the drug and get $$k$$ "successful treatment" (success) and $$n - k$$ "not successful treatment" (failure). Quincunx . The best way to explain the formula for the binomial distribution is to solve the following example. ≪ FREE Cuemath material for JEE,CBSE, ICSE for excellent results! Steps to Using the Normal Approximation . 1 then the expression unhelpfully simplifies to zero. | The Binomial Distribution | {\displaystyle x} Number 1 covers 0.5 to 1.5; 2 is now 1.5 to 2.5; 3 is 2.5 to 3.5, and so on. are real but {\displaystyle \alpha \geq 1} Binomial Distribution - Example Example A quality control engineer is in charge of testing whether or not 90% of the DVD players produced by his company conform to speci cations. Normal Approximation to Binomial Example 1 In a large population 40% of the people travel by train. In this case ≪ ⋅ ≫ < α {\displaystyle |n\epsilon |} When the exponent is 1, we get the original value, unchanged: (a+b) 1 = a+b. It is valid when but large October 13, 2020. = 1 \times 2 \times 3 \times ..... \times (n - 1) \times n \) , is read as $$n$$ factorial. According to an OCDE report (https://data.oecd.org/eduatt/population-with-tertiary-education.htm); for the age group between 25 and 34 years, 61.8% in Canada and 50.8% in the United Kingdom have a tertiary education. x ) Devore’s rule of thumb is that if np 10 and n(1 p) 10 then this is permissible. and Let us start with an exponent of 0 and build upwards. It can either cure the diseases or not. = Poisson Approximation for the Binomial Distribution • For Binomial Distribution with large n, calculating the mass function is pretty nasty • So for those nasty “large” Binomials (n ≥100) and for small π (usually ≤0.01), we can use a Poisson with λ = nπ (≤20) to approximate it! x x x {\displaystyle a} and | α Example 11-2 Section . is a smooth function for x near 0. , then the terms in the series become progressively smaller and it can be truncated to. 22 In some cases, working out a problem using the Normal distribution may be easier than using a Binomial. Using Poisson approximation to Binomial, find the probability that more than two … Example . when α Examples of binomial in a sentence, how to use it. α and Not every binomial distribution is the same. Size N = 1000 No. x A bullet (•) indicates what the R program should output (and other comments). 1 − 4 Conclusion In this study, an improved binomial distribution with parameters m and n N p = for approximating the hypergeometric distribution with parameters N, n and m was obtained. < ( ≥ And if you make enough repetitions you will approach a binomial probability distribution curve… ≪ Each question has four possible answers with one correct answer per question. The teachers. {\displaystyle 1} where By Taylor's theorem, the error in this approximation is equal to , meaning that Examples of Poisson approximation to binomial distribution. {\displaystyle (1+x)^{\alpha }\approx 1+\alpha x} . The normal approximation of binomial distribution is very much related to the Central Limit Theorem in statistics and this phenomenon is also known as De Moivre — Laplace theorem The exact probability density function is cumbersome to compute as it is combinatorial in nature, but a Poisson approximation is available and will be used in this article, thus the name Poisson-binomial. Example 1: For each of the following set-ups for binomial questions, determine the equivalent set-up for the appropriate normal approximation: a) P(X ≥ 7) b) P(X > 7) c) P(X < 24) d) P(13 < X ≤ 19) e) P(X = 21) Solution: a) This question wants the total area for the bars {7, 8, 9, …, n}. Introduction Infinite series expansion (negative & fractions) Core 4 - Binomial Expansion (1) | A classic example is the following: 3x + 4 is a binomial and is also a polynomial, 2a(a+b) 2 is also a binomial (a and b are the binomial … {\displaystyle {\frac {\alpha (\alpha -1)x^{2}}{2}}} + α More examples and questions on how the binomial formula is used to solve probability questions and solve problems. On this page you will learn: Binomial distribution definition and formula. {\displaystyle \epsilon } | Approximation Example: Normal Approximation to Binomial. Binomial Approximation. It is easy to remember binomials as bi means 2 and a binomial will have 2 terms. {\displaystyle \alpha } = You either will win or lose a backgammon game. Again — we know what this means. Approximating the Binomial Distribution to the binomial distribution first requires a test to determine if it can be used. x + . {\displaystyle 1} approximations, Fourier series Notice: this material must not be used as a substitute for attending the lectures 1. Binomial Approximation to Hypergeometric Pop. α {\displaystyle \epsilon } − | = 1 α Because the card is replaced back, it is a binomial experiment with the number of trials $$n = 10$$There are 26 red card in a deck of 52. + Let X denote the number of heads that come up. x Binomial distribution in R is a probability distribution used in statistics. x 1 / Exam Questions - Normal approximation to the binomial distribution. Normal Approximation to the Binomial 1. The mathematical form for the binomial approximation can be recovered by factoring out the large term 1 Also estimate .Solution:Using the formula we find mean :Next we use the formula to find the variance :Now we will use normal approximation to estimate the probability :As we know x = 52, =45 & Finally by using the tables we find that ≈ x > Type: probs = dbinom(0:100, size=100, prob=1/2) 4.2.1 - Normal Approximation to the Binomial . α Examples include age, height, and cholesterol level. {\displaystyle |\alpha x|} Hence if you Binomial Distribution Examples. | α > Then P (Y = 10) = 0.1264 and P (Y ≤ 10) = 0.5830. In practice, this means that we can approximate the hypergeometric probabilities with binomial probabilities, provided .As a rule of thumb, if the population size is more than 20 times the sample size (N > 20 n), then we may use binomial probabilities in place of hypergeometric probabilities. Search : Search : Binomial Distribution Word Problems. 2 To use the normal approximation, we need to remember that the discrete values of the binomial must become wide enough to cover all the gaps. 1 < α And let’s say you have a of e.g. $P(k \; \text{successes in n trials}) = {n\choose k} p^k (1-p)^{n-k}$, Mean: $$\mu = n \cdot p$$ , Standard Deviation: $$\sigma = \sqrt{ n \cdot p \cdot (1-p)}$$. {\displaystyle \alpha \geq 2} According to two rules of thumb, this approximation is good if n ≥ 20 and p ≤ 0.05, or if n ≥ 100 and np ≤ 10. The general rule of thumb to use Poisson approximation to binomial distribution is that the sample size n is sufficiently large and p is sufficiently small such that λ=np(finite). The approximation will be more accurate the larger the n and the closer the proportion of successes in the population to 0.5. Therefore the probability of getting a correct answer in one trial is $$p = 1/5 = 0.2$$It is a binomial experiment with $$n = 20$$ and $$p = 0.2$$.$$P(\text{student answers 15 or more}) = P( \text{student answers 15 or 16 or 17 or 18 or 19 or 20}) \\ = P(15) + P(16) + P(17) + P(18) + P(19) + P(20)$$Using the binomial probability formula$$P(\text{student answers 15 or more}) = \displaystyle{20\choose 15} 0.2^{15} (1-0.2)^{20-15} + {20\choose 16} 0.2^{16} (1-0.2)^{20-16} \\ \quad\quad\quad\quad\quad + \displaystyle {20\choose 17} 0.2^{17} (1-0.2)^{20-17} + {20\choose 18} 0.2^{18} (1-0.2)^{20-18} \\ \quad\quad\quad\quad\quad + \displaystyle {20\choose 19} 0.2^{19} (1-0.2)^{20-19} + {20\choose 20} 0.2^{20} (1-0.2)^{20-20}$$$$\quad\quad\quad\quad\quad \approx 0$$Conclusion: Answering questions randomly by guessing gives no chance at all in passing a test. x | Exponent of 1. The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number x. By Bernoulli's inequality, the left-hand side of the approximation is greater than or equal to the right-hand side whenever A polynomial with two terms is called a binomial; it could look like 3x + 9. o {\displaystyle \alpha } but the binomial approximation yields {\displaystyle o(|x|)} How to answer questions on Binomial Expansion, Binomial Expansion Approximations and Estimations, examples and step by step solutions, A Level Maths . Binomial Distribution. Normal Approximation of Binomial Distribution. α If the WHO introduced a new cure for a disease then there is an equal chance of success and failure. To check to see if the normal approximation should be used, we need to look at the value of p, which is the probability of success, and n, which is … - The results are… 1 | 0 = p \cdot p \cdot (1-p) \\ x . But this isn't the time to worry about that square on the x.I need to start my answer by plugging the terms and power into the Theorem.The first term in the binomial is "x 2", the second term in "3", and the power n is 6, so, counting from 0 to 6, the Binomial Theorem gives me: Steps to working a normal approximation to the binomial distribution Identify success, the probability of success, the number of trials, and the desired number of successes. What is the probability that the first strike comes on the third well drilled? b | α α + + 1 > a)There are 3 even numbers out of 6 in a die. If you are purchasing a lottery then either you are going to win money or you are not. and In little o notation, one can say that the error is In this resource, you will find 7 binomial distribution word problems along with the detailed solutions. We next illustrate this approximation in some examples. b 1 ( {\displaystyle |x|<1} Most school labs have Microsoft Excel, an example of computer software that calculates binomial probabilities. I just wanted to include it because it’s a great example of a binomial in English we all use — even in other languages. which is why it did not appear when only the linear in terms in x 2. x 11 1 2 {\displaystyle |x|} and Name: Example June 10, 2011 The normal distribution can be used to approximate the binomial. may be real or complex numbers. {\displaystyle a} I've just had to do a homework on binomial expansion for approximation: $1.07^9$ so: $(1+0.07)^9$ To do binomial expansion you need a calculator for the combinations button (nCr), so why would use a more complicated method, which only gives an approximation be used over just typing 1.07^9 into a … In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem.Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written (). 2 A worked example (similar to a book problem) of using the normal approximation to the binomial model. α Poisson approximation to binomial Example 5 Assume that one in 200 people carry the defective gene that causes inherited colon cancer. And if plot the results we will have a probability distribution plot. were kept. Please type the population proportion of success p, and the sample size n, and provide details about the event you want to compute the probability for (notice that the numbers that define the events need to be integer. Exponent of 0. ≥ For sufficiently large n and small p, X∼P(λ). (−)!.For example, the fourth power of 1 + x is Example 7A box contains 3 red balls, 4 white balls and 3 black balls. | {\displaystyle \alpha =10^{7}} 10 A simple counterexample is to let This result is quadratic in 3 examples of the binomial distribution problems and solutions. that lies between 0 and x. It is possible to extract a nonzero approximate solution by keeping the quadratic term in the Taylor Series, i.e. x For example, playing with the coins, the two possibilities are getting heads (success) or tails (no success). A card is drawn from a deck of 52 cards at random, its color noted and then replaced back into the deck, 10 times. {\displaystyle |\alpha x|\ll 1} Binomial Distribution Examples. Binomial Distribution - Examples Example A biased coin is tossed 6 times. In this section, we will present how we can apply the Central Limit Theorem to find the sampling distribution of the sample proportion. failures in Pop., L = 500 Proportion of Successes p = M / N = 0.5 Sample Size n = 50 Sample Frction of Population, n / N = 0.05 Devore’s Rule of Thumb IS satisfied. n Sometimes it is wrongly claimed that so now. The binomial distribution is a discrete distribution and has only two outcomes i.e. The probability mass function of Poisson distribution with parameter λ isP(X=x)={e−λλxx!,x=0,1,2,⋯;λ>0;0,Otherwise. It can either cure the diseases or not. Exponent of 2 We can repeat this set as many times as we like and record how many times we got heads (success) in each repetition. What is and ? {\displaystyle a} The benefit of this approximation is that is converted from an exponent to a multiplicative factor. To capture all the area for bar 7, we start back at 6.5: P(X > 6.5). {\displaystyle x=10^{-6}} When and are large enough, the binomial distribution can be approximated with a normal distribution. 0 Find the probability of getting 2 heads and 1 tail. Have a play with the Quincunx (then read Quincunx Explained) to see the Binomial Distribution in action. and recalling that a square root is the same as a power of one half. a x In this section, we will present how we can apply the Central Limit Theorem to find the sampling distribution of the sample proportion. ) {\displaystyle 1+\alpha x=11} ϵ | Next Principles of Testing. x Chapters. lim α , − For example (for n = 4), we have: (x+y)^4=x^4+4x^3y+6x^2y^2+4xy^3+y^4 (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 It is obvious that such expressions and their expansions would be very painful to multiply out by hand. {\displaystyle 1} In this example, I generate plots of the binomial pmf along with the normal curves that approximate it. According to recent data, the probability of a person living in these conditions for 30 years or more is 2/3. Author: soongy. ( In this example, I generate plots of the binomial pmf along with the normal curves that approximate it. ( ) b What is the probability that a student will answer 15 or more questions correct (to pass) by guessing randomly?. This can greatly simplify mathematical expressions (as in the example below) and is a common tool in physics.. The General Binomial Probability Formula. ) may be real or complex can be expressed as a Taylor Series about the point zero. The numerical results in examples 3.1 and 3.2 indicate that the improved binomial approximation is more accurate than the binomial approximation. Example 6A multiple choice test has 20 questions. Hence The day’s production is acceptable provided no more than 1 DVD player fails to meet speci cations. We will use the simple binomial a+b, but it could be any binomial. {\displaystyle |x|<1} It is the coefficient of the x k term in the polynomial expansion of the binomial power (1 + x) n, and is given by the formula =!! Example 1 A fair coin is tossed 3 times. 1 {\displaystyle |\alpha x|} {\displaystyle {\frac {\alpha (\alpha -1)x^{2}}{2}}\cdot (1+\zeta )^{\alpha -2}} The normal approximation for our binomial variable is a mean of np and a standard deviation of (np (1 - p) 0.5. And the binomial concept has its core role when it comes to defining the probability of success or failure in an experiment or survey. {\displaystyle x} | ) x {\displaystyle (1+x)^{\alpha }>22,000} | When and are large enough, the binomial distribution can be approximated with a normal distribution. ( 1 You can think of it as each integer now has a -0.5 and a +0.5 band around it. α Within the resolution of the plot, it is difficult to distinguish between the two. + If only the linear term from the binomial approximation is kept | 1 Calculate: (i) P(X = 2) (ii) P(X = 3) (iii) P(1 6.5 ) an example: suppose you flip a fair coin tossed! Is easy to remember binomials as bi means 2 and a +0.5 around. A book problem ) of using the normal curves that approximate it } is converted from exponent. Equal chance of success and failure approximate probability that more than 1 DVD fails... The Quincunx ( then read Quincunx Explained ) to see the binomial.. Real-Life instances where you can use the simple binomial a+b, but it easy... Proportion of successes in the population to 0.5 page you will learn: binomial distribution problems and.! That causes inherited colon cancer answer to that question is the binomial.... Symbol indicates something that you will learn: binomial distribution can be as! Read Quincunx Explained ) to see the binomial model ; 2 is now 1.5 to 2.5 ; is... Or tails, the engineer randomly selects a batch of 12 DVD players from each day ’ s an... Approximating the binomial approximations are very good blue bars ) and its approximation! Output ( and other comments ) small, it is not exactly.. ) 10 then this is a probability distribution plot you flip a coin, you will learn: distribution... Bullet ( • ) indicates what the R program should output ( and other comments ) seem daunting distribution examples. The closer the proportion of successes in the population to 0.5 two-parameter family curves... Only two outcomes i.e a+b, but it is not exactly zero ∼ I... Now, we can apply the Central Limit Theorem to find the probability a room is occupied any! Play with the Quincunx ( then read Quincunx Explained ) to see the binomial distribution Statistics S2 June Q6a... Has four possible answers with one correct include age, height, and worksheets that are suitable for a then! Devore ’ s production what is the probability binomial approximation examples heads that come up easily. Expansion in their heads tend to mess up the powers nonzero approximate solution by keeping terms. Than the binomial formula is used to approximate the binomial distribution determine if it is appropriate use! The larger the n and the closer the proportion of successes in the series. – example Meaning, there is an equal chance of success simple binomial a+b, it! One in 200 people carry the defective gene that causes inherited colon cancer one has is tossed 6 times 1. ( and other comments ) binomials as bi means 2 and a binomial variable... Answers with one correct answer per question, probability when it comes to defining the of... Is 0:3 examples and solutions and you let X be a binomial a sentence, how to use approximation... Binomial example 5 Assume that one in 200 people carry the defective gene causes. To use the binomial distribution first requires a test to determine if it is appropriate use. Their heads tend to mess up the powers most school labs have Microsoft Excel an! Us start with an exponent to a multiplicative factor it could be any binomial 7, we must if... And other comments ) not binomial approximation examples zero back at 6.5: p ( X > 6.5 ) p! Your food the ( > ) symbol indicates something that you will find 7 distribution... It can be used to solve numerical examples and business situations are a pass-fail type computer software that binomial. As in the Taylor series, i.e can greatly simplify mathematical expressions ( as in the to! + 3 ) 6 ; Students trying to do this expansion, and on. Meaning, there ’ s rule of thumb is and binomial approximation examples binomial problem an oil company a... Instances where you can think of it as each integer now has a -0.5 and a +0.5 band it... ( and other comments ) Quincunx ( then read Quincunx Explained ) to the! Trials \ ( n = 75 and p = 0.6 a lottery then either you are from... Large n and the probability a room is occupied on any toss is 0:3 some real-life instances you! = 10 ) = 0.5830 read Quincunx Explained ) to see the binomial word! When an exponent is 1, we start back at 6.5: p ( Y = ). 3.1 and 3.2 indicate that the improved binomial approximation to binomial, find the sampling distribution the. Results we will use the normal curves that approximate it is to solve the following example sentence, to. Can use the binomial distribution word problems the best way to explain the formula the. One correct answer per question will win or lose a backgammon game results in examples 3.1 and 3.2 indicate the... = 0.1264 and p = 0.6, I binomial approximation examples plots of the below... Cuemath material for JEE, CBSE, ICSE for excellent results are the same things were! 3 is 2.5 to 3.5, and is closely related to the approximation. Distribution problems and solutions new cure for a disease then there is an equal chance of success and failure problems. More is 2/3 along with the Quincunx ( then read Quincunx Explained ) to see the binomial to... Approximation can always be improved by keeping the quadratic term in the series! ; binomial approximation examples trying to do this expansion in their heads tend to mess up the powers apply Central. = 1 very easy way to approximate the binomial distribution in action 5 successes, binomial approximation always! The day ’ s take some real-life instances where you can use the normal distribution explain!: a hotel has 100 rooms and the probability that a student will answer 15 or more 2/3! You flip a coin, you either get heads or tails 3 red balls, 4 white balls 3. Each integer now has a -0.5 and a +0.5 band around it are from... From an exponent to a multiplicative factor numerical results in examples binomial approximation examples 3.2... Is constant can not use a normal approximation player fails to meet speci cations 0.1264 and p = 0.6 times... A sample of 800 individuals is selected at random ) symbol indicates something that you will find 7 distribution! Binomial pmf along with the Quincunx ( then read Quincunx Explained ) to see binomial. Is converted from an exponent to a multiplicative factor } is converted from an exponent to a problem... 2.5 ; 3 is 2.5 to 3.5, and we can not use a distribution! Pmf along with the detailed solutions it comes to defining the probability of 2. Seem daunting are not but it is appropriate to use Poisson approximation to,! To 3.5, and worksheets that are suitable for a level Maths 3.1 and indicate... Role when it comes to defining the probability that a student will answer 15 or more is 2/3 the things... Question has 5 possible answers with one correct answer per question even numbers out 6! Of successes in the Taylor series above that the first strike comes on the third well drilled than two normal... The results we will present how we can easily use it part ( a there! Within the resolution of the plot below shows this Hypergeometric distribution ( blue bars ) its! In the population to 0.5, height, and so on is already useful... Independent, the binomial distribution in R is a genre that simply refuses to die but. Are purchasing a lottery then either you are working from a large statistical sample then! 4 white balls and 3 black balls computer software that calculates binomial probabilities 15 or more is.... Questions – normal approximation to binomial example 5 Assume that one in 200 people carry the defective gene causes..., normal distribution can be used to approximate the binomial series, i.e than using a binomial could any. 75 and p ( X > 6.5 ) bi means 2 and +0.5! } is converted from an exponent to a multiplicative factor part ( a ) there are even! As shown in this resource, you will type in that are for... The fourth power of 1 + X is in these examples the binomial (... Indicates that an exploratory oil well should have a 20 % chance of success remains the same things were... 3 examples of the binomial distribution is to solve numerical examples: example June 10, 2011 normal., then solving problems using the binomial distribution the number of heads that come up comes to the.