Poisson distribution - Example 1

The following problem is solved:
At a Major League Baseball park, five customers arrive at a concession stand on average in a ten-minute period.
A) What is the probability that in a ten-minute period exactly three people will arrive at the concession stand?
B) Find the probability of exactly 3 customers arriving in 20 minutes?
C) Find the probability of 3 or fewer customers arriving in 20 minutes?
D) Find the probability of more than 3 customers arriving in 20 minutes?

[@aftikharmominzada1912]

If all instructors could make it this concise and to the point! Thank you for making this video.

[@aplusintegral]

I’m glad that you found it helpful. Please share with your friends. And do not hesitate to ask me for a video about a problem you would like to see the solution.

[@ismaelnewman1382]

All of these, the videos and the questions were great help Thank you

[@aplusintegral]

Thanks Ismail. Good to know this is helpful

[@adrianmonk4468]

Thanks for asking this question Maluleka, and thanks for answering it professor. Now I can solve most of the problems in my textbook. However, what happens if the interval of the question is smaller than the interval in the information.

[@aplusintegral]

Thanks Adrian, You can see this other video kzbin.info/www/bejne/b2jIgH56lMSFjpY, where I show an example in which the interval in the question is smaller (1/10) than the interval in the information of the problem.

[@melanieclaire509]

Thank you very much all the way from Cameroon

[@aplusintegral]

I’m really glad you found it useful please share with your friends. If you have any math, probability or statistics exercise that you want to be featured in this channel, please let us know.

[@tranquillamp2298]

Why is t=1 and not 10 because it is 10mins for the first question

[@aplusintegral]

Hi Nhlayiseko, Thanks for your interest. The first thing you need to notice is that t is not the time in minutes. In a poison distribution problem there are always two important intervals. One is in the information of the question that gives the average of the variable. For example. here the average that they gives us is 5 customers in a 10 minutes interval. So. 10 minutes is the information interval. There is also another interval stated in the problem, and this is always in the question. For example in question a) the question interval is 10 minutes again (they asked for the probability in a 10 minute interval). t is the number of information intervals in the question interval. In question a) we notice that the interval in the question is exactly the same interval of the information, then t=1. In Question b) the question is in a 20 minutes interval, here there are 2 intervals of 10 minutes, then for question b) t= 2.

[@sobahlemaseko9189]

@@aplusintegral wow I get it now thanks

[@sensationalgoodluckkirika8972]

I also gat it now 👍

[@srinivassr3826]

@@aplusintegral Thanks for time❤️

[@jagathrajapaksha9093]

great explanation, thank you so much, specially, I understood the difference between mu and lambda

[@aplusintegral]

You are really welcome! Yes, I have followed the notation of some textbooks (Groebner's textbook from Pearson for example) that show the difference between the average (lambda) in the segment of the information of the question, and the average (mu) in the interval of the actual question of the problem. Some other classical textbooks only consider in their explanation the average in the actual question of the problem, and for them, mu and lambda are the same thing. So we need to be careful what type of notation your book is using. Thanks a lot for your participation.

[@jagathrajapaksha9093]

Thank you so much for your time giving further information. good day Sir @@aplusintegral

[@favoritetroll9774]

In part D, why did you divide the probability into p(x=4), p(x=5) , p(x=6) , p(x=7)...etc. Isn't in the main question only 5 customers arriving so wouldn't it make sense to do it only till x=4 and x=5. idk how u got x=6 and upwards

[@aplusintegral]

Hello there, Thanks for your very nice question! Observe that the problem say, In average 5 customers. That does not mean that are always arriving 5 customers. If the average is five, the number of customers could be 1 or 2, or 4, or 5, or 6, or 7, or whatever other number. When somebody compute the average (the mean), the answer is 5, but 5 not always happens. Every good poisson distribution problem must give the average of the random variable, but not the maximum of the value. Please let me know if this is clear now, or you need more explanations.

[@favoritetroll9774]

@@aplusintegral Thank that makes it a bit but can you please solve this question as I have no idea how it works.

[@favoritetroll9774]

Two hundred passengers have made reservations for an airplane flight. If the probability that a passenger who has a reservation will not show up in 0.01, what is the probability exactly three will not show up?

[@aplusintegral]

@@favoritetroll9774 Thanks again for your question. Use the binomial distributionfor this problem. Why? I have solved a similar problem but with two questions and only 10 passengers, but the solution is similar. check it and you will notice. The problem is in kzbin.info/www/bejne/bJ2sdqOBYpagkJo.

[@aplusintegral]

However is you want a solution for this in particular. Please notice that this is a problem in which we know the exact number of trials n, which in this case is 200. That is because in this case we want to know of these 200 passengers how many of them will show up. In this case a binomial distribution will be the best model for getting the right question the formula is P(x) =nCx(P^x)(1-p)^(n-x) (I use the symbol ^ for the power, and nCx for the combinations of x object chosen from a set of n objects)), In your problem n=200 and p, the probability of no-show up is 0.01. Because in this case we need to find the probability of exactly 3 passengers do not show up, x=3 an then by substituting in the formula, we have P(3) =200C3(0.01^3)(0.99^197) You can use your calculator for doing this and notice that rounded two four decimals this is=0.1814

[@trouble.nyirongo4483]

This is awesome you are a great recture

[@aplusintegral]

Thanks. I am Hernando. If you have further questions do not hesitate and type them in the comments. I’m looking for students questions for making videos of interest. A little about me can be found in the Wikipedia page en.m.wikipedia.org/wiki/Hernando_Burgos-Soto

[@aplusintegral]

Thanks Adrian, You can see this other video kzbin.info/www/bejne/b2jIgH56lMSFjpY, where I show an example in which the interval in the question is smaller (1/10) than the interval in the information of the problem.

[@veronicamiles7823]

Thanks so much. Do you do statistics tutoring as well?

[@aplusintegral]

I am very glad that you found this video useful. Unfortunately, I do not tutor at this time. However, if you make a comment asking me a math question, I'll be happy to make a video of the answer and post it here in this channel.

[@michaelitam3132]

Great teaching skills. Much appreciated!

[@aplusintegral]

I'm really happy knowing you found it useful!

[@maxmorris4562]

Thanks so much that's really clear, you made it easy

[@aplusintegral]

I happy to be of help. Let me know if you need help with the solution of any other exercise, I enjoy doing this.

[@MM-ys4fx]

The formula for the crash multiple is: ∞/(∞-X), X is randomly selected from 0~∞. The data are 6.51 and 9.51 The result is = 33.88 Can you explain to me how to find this result please Can you give me a solution to this issue

[@aplusintegral]

Hello MM-ys4fx, Could you provide more information about the context of your question. I will be very happy helping you and providing a video explaining. However, I will need more information. What is the topic? What do you mean when you say "crash multiple"? What is the subject you are studying?

[@MM-ys4fx]

@@aplusintegral I am not good at English but I am a moment that the responsibility is giving a number and predicting the result The Data Are 6.51 and 9.51 The Result Is = 33.88 The result is the number of prediction I want it because I discovered that one of the games are going on this pattern

[@aplusintegral]

@@MM-ys4fx I still want to help you. however I do not understand exactly what Is the actual question. If you want, you can write the whole question in your mother tongue ( if is taking from a book, type exactly as it is), and I will try to translate using a digital translator.

[@MM-ys4fx]

@@aplusintegral شكرا لك حقا انك انسان رائع ما اقصده اني العب لعبه اسمها الطيارة وهذه الطيارة تتحطم في ارقام مميزه وكأنها تستخدم معادلة معينة او خورازميه معينة ولذالك بحثت حتى حصلت على هذه المعادلة التي تستخدمها اللعبة

[@aplusintegral]

@@MM-ys4fx Great I like the problem, and I think I can help you to solve it. But Still I have some questions: The plane can crash in any real number, from minus infinite to infinite with the same probability? You told me that there are two data 6.51 and 9.51. What role do these two numbers play?

[@khinayemyint2148]

Thank you Sir,I really clear .

[@aplusintegral]

You’re welcome!

[@nuguyaguuu]

Hi sir, ur video makes me so motivate to learn this material. But can I ask, why when we find question c we should put 1 in cumulative part in excel? Thank u

[@aplusintegral]

Hi Alifia, Thanks for your words. Any time that we want to compute a probability of the form, P(X

[@the_new_rolo2057]

Thank you for also explained it in excel

[@aplusintegral]

I’m glad to know that you found it useful

[@yusufmostafa5404]

I loved it

[@aplusintegral]

I’m glad you found it useful, please let me know if there is any other problem you would like to be featured in this Channel

[@chetangarg01]

Sir help me to solve this problem Arrivals of customers at a telephone booth follow poisson distribution, with an average time of 10 minutes between one arrival and the next. The length of the phone call is assumed to be distributed exponentially with a mean of 3 minutes. Find a. The average number of persons waiting and making telephone calls b. The average length of the queue that is formed from time to time. c. Probability that a customer arrive and find telephone booth is busy. d. Probability that a customer arrive and find telephone booth is empty. e. The average time spent by a customer in telephone booth

[@aplusintegral]

Hi Chetan, the last question (e), the average time spent by the customer is given in the information of the question. That is the mean of the exponential distribution (3 minutes is the average of the phone call). For the other questions I made a video. This is in kzbin.info/www/bejne/mIikdIN9jrdmmtU. I hope you can find it useful

[@chetangarg01]

@@aplusintegral thanks for solving a problem for me and creating a seperate video. It could help me for project.

[@amermuzaffar8526]

thank you sir.

[@aplusintegral]

I‘m glad you found it useful.

[@kyuutiful]

Hi sir how did you got P(X < 3) = 1 (P

[@aplusintegral]

Hi Yuyu, Thanks for your question. This problem is solved using the probability of the complement of an event P(A)=1-P(complement of A). When you ask P(x>3), 3 is not included, so you need to include it in the complement. In case of a discrete variable, as a poisson variable, to be more than 3 is actually to be or 4, or 5 or,6, etc. 3 is not included. Then if you are going to use the complement, this will take into account all the numbers that are not there: 0, 1, 2 and 3. For that reason you compute 1- P(x=

[@kyuutiful]

@@aplusintegral thank you Sir 🙏🏻

[@braveman-qm6qn]

awesome you gained a sub

[@aplusintegral]

😋I m glad you’ve found it useful

[@abdulaiabdul-latif6835]

Super tutorial

[@aplusintegral]

Glad you think so! You can write in the comments any math or statistics problem you want to get an explanation.

[@markevans5731]

thank bro, good video

[@aplusintegral]

I’m glad you found it useful!

[@nanditairawan3344]

How e^-5?

[@aplusintegral]

Thanks for your question. I understand that your question is how to compute e^-5. Well e=2.7182818, approximately. So, one way to compute this will be using this approximation in your calculator and use minus 5 as the exponent. However, every Financial and Scientific calculator has a key for the exponential function (e^x) you can use this function of your calculator. If you want to know more about the number e, you can see my video: kzbin.info/www/bejne/bITLfYFud52retU

[@tanyajain7802]

Sir please give instructions clearly

[@aplusintegral]

I would like to be of help. Please tell us about what you need clarification.

[@mujuniandrew8565]

How t=1yet it’s ten minutes

[@aplusintegral]

Thanks for your interest Mujuni. The first thing you need to notice is that t is not the time in minutes. In a poison distribution problem there are always two important intervals. One is in the information of the question that gives the average of the variable. For example. here the average that they gives us is 5 customers in a 10 minutes interval. So. 10 minutes is the information interval. There is also another interval in the stated in the problem, and this is always in the question. For example in question a) the question interval is 10 minutes again (they asked for the probability in a 10 minute interval). t is the number of information intervals in the question interval. In question a) we notice that the interval in the question is exactly the same interval of the information, then t=1. In Question b) the question is in a 20 minutes interval, here there are 2 intervals of 10 minutes, then for question b) t= 2.

[@kyuutiful]

@@aplusintegral then if in a five-minute period, t = 1 ?

[@aplusintegral]

@@kyuutiful Thanks for your question Yuyu. Please notice that they give you two intervals. Here, in the information of the problem, they tell you average in 10 minute period. Then they tell you another interval in the actual question (When they ask you find the probability in ....?) Is this last interval is again 10, then t=1, if this last interval is 20 then t=2, because there are two information intervals in the question interval. If the question interval is 30. then t=3, etc. Yuyu, If you have a similar question, please write here, and I will tell you the value of t, for your question

[@kyuutiful]

@@aplusintegral thank you Sir 🙏🏻

[@bashirumahmud3381]

My problem is that I don't know how to use the formula in the calculator to arrive at the final answer.

[@aplusintegral]

Hi Bashiru, Thanks for your interest. I understand your problem. Every calculator is different but every scientific or financial calculator has a key for the exponential function. You can see the following video in which I explain how to use that ba II + calculator of Texas instrument for solving a similar problem: kzbin.info/www/bejne/hKXQc6V6r9R-iKs. This is a long video, but the explanation of the calculator is from about minute 18

[@Aditya.259]

What is the value of e

[@aplusintegral]

e is approximately 2.7182818285. It is la base of natural logarithms. Every scientific calculator has a function to compute its powers, e^x. In this channel there is a video showing deduced to how this constant can be obtained here it is the link kzbin.info/www/bejne/bITLfYFud52retU

[@Aditya.259]

@@aplusintegral thankyou 👍👍

[@abdulaiabdul-latif6835]

Great

[@aplusintegral]

Glad you found it useful!

[@mindyu-uz3cx]

how did you got the 3 on the x? is it 5-2?

[@aplusintegral]

Hi Chae, thanks for your question. Because here the question is “what is the probability of exactly 3 customers? Then x=3. If another question is ‘“what is the probability of exactly 7, then x=7, and so on. The x depends on what is the value in the question

[@mujuniandrew8565]

What’s e

[@aplusintegral]

e is a constant in mathematics. It is the famous base of the natural logarithm. A good approximation to this number is e=2.71828182846. However it will be better if you look at your calculator and check for the key that has the function e^x. Every scientific calculator has this key. It will be useful for computing this type of results.