Binomial Distribution

Need help with your homework? Look no further! Our subject experts are ready to effortlessly handle your assignments, so you can finally say goodbye to stress and hello to top grades.

Please enable JavaScript in your browser to complete this form.
Click or drag files to this area to upload. You can upload up to 3 files.
Get a response in under 15 min

Let us assume a variable X that carries two values 1 and 0 consisting a probability p and q respectively, whereq = 1 – p and this arrangement is popularly known as Bernoulli distribution. Provided,

p = the probability of success

q = called the probability of failure.

Thus, in the case ofn number of trials, probability ‘X’attempts to (x≤n) in the Binomial distribution being provided to us. So the probability mass function will be as follows:

P [X =x] = (nx) px .qn-x, x= 0,1,….,n

Where, n = Number of free trials

x = Number of successful attempts

p = Probability of successful attempts of the trial

q= 1- p

(nx) = n cx

Usually, it is represented as B(n, p).

 Properties:

  • Σnx=0 P [X =x] = 1
  • Distribution function

F(x) = P [ X≤ x] = Σxk=0 (nk) pk (1-p)n-k

  • The first two moments about the source

µ’1 = Σnx=0. X (nx) px.qn-x

= np. Σnx=1 (n-1x-1) px-1..qn-x

= np.(p+q)n-1= np

µ’2 = Σnx=0. X2 (nx) px.qn-x

= Σnx=0. [ x (x-1) +x) (nx) px.qn-x

= Σnx=0. X(x-1) (nx) px.qn-x  + np

= n(n-1) p2Σnx=2. (n-2x-2) px-2.qn-x +np

= n(n -1) p2. (q+p)n-2 + np

= n(n- 1) p2 + np.

So,               µ1=µ1=np…….which is the mean

µ2=µ2-(µ1)2

=n( n-1)p2 + np – n2p2

=np-np2

=np(1-p)……… is the variance.

Thus we acquire µ3 and µ4 in a similar manner

  • Skewness:

β1 = (1- 2p)2/npq,        ɣ1 = 1-2p/ √nq

 

  • Kurtosis :

Β2 = 3 + 1- 6pq/ npq,      ɣ1= 1-6pq/ npq

(vi) Mode is nothing the value that x consists for which P[X = n] is maximum.

When (n + 1) p is not an integer,

Mode = Integral part of (n + 1)p.

When (n + 1) p is an integer, we obtain two modes

Mode = (n + 1) p and (n + 1) p – 1. ·

 

Example 1. Find out the binomial distribution which has 8 as its mean and variance 4. Also calculate the mode.

Solution. We know np = 8 and npq = 4

On dividing we obtain q = ½                              =>        p =  l – q  = 1/2

Also n = 8/p = 16

Thus the required binomial is B(16,1/2).

Now, (n + 1) p = (16 + 1) ½=17/2=8+1/2 which infers that mode= 8.

 

Example 2. In a given binomial distribution the mean and variance are 5 and 2 respectively. Calculate P[X -1].

Solution.We know np = 5,  and  npq = 5/2

After dividing we obtain q = 1/2,           p = 1/2

Also,                            n = 5/2 = 10

P[X – 1] = P[X = 0] + P[X = 1]

= (100) p0.q10 + (101) p1.q9

= (1/2)10 + 10.(1/2).(1/2)9 = 11/1024 = 0.01

 

Example 3.In a shooting contest, a man probably hits the target 215 times. Now if he fires 5 times, find out the probability of hitting the target (i) at least twice (ii) at most twce.

Solution. Let p = hitting a target = 2/5. q = 1 – p =3/5. n = 5

(i)  P[at least twice hitting] = 1 – [P(no hitting) + P(one hitting)]

=1 – [(50)p0 q5 + (51)p1 q4

= 1 – [(3/5)5 + 5. (2/5).(3/5)4]

=1 – 0.337 = 0.66

(ii) P[ at most twice hitting)= P(no hitting) + P( one hitting) + P(two hitting)

= (50)p0 q5 + (51)p q4 + (52)p2 q3

= (3/5)5 + 5. (2/5).(3/5)4] + 10.(2/5)2 (3/5)3

=0.68.

 

Example 4. If 4 scooterists among 12 does not have driving license, then find out the probability of a traffic inspector to randomly select 4 scooterists:

(i) for not keeping driving license.

(ii) at least 2 for not keeping driving license.

 

Solution.  Let, P= Probability that a scooterist does not have driving license=4/12=1/3

Then,   q = 1 – p = 2/3, n = 4

  • P (catching one scooterist having no driving license)

= (41)p1 q3

= 4. 1/3. (2/3)3 = 32/81

  • P (catching at least two scooterists having no driving license)

=1 – [P(all having license) + P(l having no license)]

= 1 – [(04) p0 q4 + (41)p q3

= 1 – [(2/3)4 + 4. (1/3).(2/3)3]

=  1 – 48/81 = 33/81 = 11/27

 

Example 5. Solve the binomial distribution represented below:

X 0 1 2 3 4 5
f 27 14 6 3 0 0

 

 

Solution.   Mean=∑ x f / ∑ f= 35/50 , n=5

Therefore,        np =35/ 50  =>  p 35/250  =0.14   and   q=  0.86

Thus the expected frequencies of the supplied binomial distribution can be evaluated from

50 (0.86 + 0.14)5

x 0 1 2 3 4 5
Expected F 24 19 6 1 0 0

 

 

PROBLEMS

  1. What is the probability of a specific student to get three prizes at a time if the 5 prizes are being distributed among 20 students?
  2. Some study shows an intersection where there are 25% right turns and zero left turns. Then calculate the probability of one out of the next four vehicles turning right.
  3. 6 and 2 are the mean and variance of a binomial distribution respectively. Hence calculate P[X > 1],P[X = 2].
  4. In a binomial distribution that consists of 5 independent trials the probability of getting I and 2 are 0.4096 and 0.2048. Then calculate the parameter p of the arrangement.
  5. An experiment is performed 4 times, twice as often as it fails and succeeds. Then find out the probability that in next five trials there will be (i) three successes, (ii) at least three successes?
  6. A quality control engineer checks a random 3 calculators randomly from each bundle of 20 calculator. Now if each bundle contains 4 slight defective calculators. Then find out the probabilities that the inspector’s sample will comprise
  • no slight defective calculators,
  • one slight defective calculators,
  • At least two slight defective calculators.

 

  1. Fit a binomial distribution to the following distribution
x 0 1 2 3 4 5
f 3 12 21 30 25 9

 

  1. Fit a binomial distribution to the following data.
x 0 1 2 3 4
f 15 12 10 8 5

 

  1. How many times a coin must be tossed to increase the probability of getting at least one head at 87.5%?
  2. 20% defective bolts is approximately produced by a machine. A batch that is accepted if a sample of five (5) bolts taken from the same batch that contains no defective. The rejected sample contains 3(three) or more defectives. In other cases, a second sample is taken. What is the probability that the second sample is required?
  3. If probability of a defective bolt is just 0.1, find (i) mean, (ii) variance, (iii) moment coefficient of skewness and, (iv) Kurtosis for distribution of defective bolts that is in total of 400.
  4. If on an average one vessel in every ten(10) is wrecked, find the probability that out of five (5) vessels that are expected to arrive, at least four (4) will arrive safely.

ANSWERS

  1.  0.088                                      2. 0.56                           3. 0.999, 0.007
  1. p =1/5                                       5.         (i) 80/243                    (ii)192/243
  2. (i)64/125            (ii) 48/125                   (iii)13/125

7.

  1. 3 tosses are required

 

  1. 0.6144

 

  1. (i) 40,  (ii) 36,

(iii) 15′             (iv)1800

  1. 45927/50000

Links of Previous Main Topic:-

Links of Next Statistics Topics:-

Homework Blues?

Get expert help with homework for all subjects.

  • NPlagiarism-free work
  • NHonest Pricing
  • NMoney-back guarantee

Latest Reviews

Solved Sample Works

Accounting Homework

Corporate Accounting Sample

Biology Homework

Genetics Assignment Sample

Essay Writing Help

Business Plan Sample

Homework Help FAQs

Our Answers to Your Questions

How do I submit my homework?
K
L

Getting homework help is very simple with us. Students can either send us the homework via email or they can upload it to our online form here. For a quicker response, You can also chat with us at WhatsApp and submit homework directly. You are sure to get a response from our side within 10 minutes.

How much will my homework cost?
K
L

The cost of paying someone to do your homework varies depending on the service and the type of assignment. We have listed our standard pricing plans for popularly used writing services. For other kind of assignments, You can get a free instant quote from us using our online form.

We also accept partial payment to start working on your assignment help. You can pay the remaining amount when your task gets completed. No pressure of up-front payment. No hidden order costs.

Can I receive help with my homework anytime?
K
L

Yes, you can receive help with your homework anytime with us. Our online homework help services are available 24/7, allowing you to receive assistance with your homework anytime, anywhere.

For urgent homework requests, reach out to us through our LiveChat or WhatsApp channels and one of our friendly support agents will assist you in finding the right expert for your online homework help request immediately. With our services, you can rely on 24/7 availability and meeting deadlines.

Are online homework websites budget-friendly for students like me?
K
L

Yes, Our Online Homework Help websites are an affordable solution for you as a student. Compared to traditional tutoring services, MyHomeworkHelp prices their homework help services honestly and within the budget of college students. This makes it easier for you to receive assistance with your homework without breaking the bank.

What is your plagiarism-free policy?
K
L

At myhomeworkhelp, we take plagiarism very seriously and ensure that all solutions provided by our tutors are original and authentic. Our tutors are trained to provide custom-made solutions, tailored specifically to meet the requirements of each student. We do not provide pre-written papers. All our homeswork solutions are made from scratch, guaranteeing 100% orignal homework answers.

Additionally, we have strict plagiarism-detection tools in place to check all submissions for authenticity.

Is using an Online Homework Help Service cheating?
K
L

Using online homework help services is not equivalent to cheating. Our services are intended to support students with their homework and provide them with the resources they need to succeed academically. With the help of our online homework help services, students can receive immediate assistance with their homework from any location, at any time.

At myhomeworkhelp, we are committed to promoting academic integrity. Our tutors provide solutions that serve as guides for drafting your own work. It is not acceptable to submit someone else's work as your own, as this constitutes academic plagiarism.

Can I chat with my tutor?
K
L

Using our secure chat board, you can now chat directly with your assigned tutor. The chats are encrypted both ways to secure your privacy. This makes your contact with the tutor directly & confidentially, so you can better explain any requirements or changes if needed or just need updates.

You can't contact the experts outside of chat board platform. Sharing any personal information, including but not limited to contact information, goes against our Terms and Conditions and therefore may result in permanently blocking you from the platform. We take any personal data very seriously and we do it for the safety of our users.

Know more about chat board here.

What is your money-back guarantee policy?
K
L

It’s worth noting that our online homework help service rarely leads to disappointment among students. Our expert tutors, along with our support and quality assurance team, are dedicated to providing the best possible experience for our customers. However, if for any reason a student is unsatisfied with their homework help solution, we offer unlimited revisions until they are fully satisfied.

In the rare event that a student remains unsatisfied even after revisions, we offer a money-back guarantee. We want all of our students to feel confident and secure when they turn to us for assistance with their homework, and this guarantee is just one way that we demonstrate our commitment to providing the best possible service. If you have any concerns about our services or the quality of the work you receive, please contact us for support.

What is the expertise of the tutor assigned to do my homework?
K
L

At myhomeworkhelp, we take pride in our team of qualified and experienced tutors. All of our tutors undergo a rigorous selection process and are required to have a minimum of a master's degree in their respective fields. Additionally, they must pass a series of tests to demonstrate their proficiency and ability to deliver quality work. We believe in transparency and providing our clients with the best possible service. You can be confident in the expertise of the tutor assigned to do your homework.

What about privacy & confidentiality?
K
L

Using My Homework Help is absolutely safe. We care about your security, therefore we encrypt all personal data to make every user feel safe while using our services and we don’t share any personal information with any third parties without your permission. Your credit card information is not stored anywhere at My Homework Help, and use of PayPal relies on their secure payment networks. Your identity, payment and homework are in safe hands. You can always be certain of getting professional help and remaining anonymous, while using My Homework Help.