# Poisson Distribution vs Binomial Distribution

I have a specific problem here. It is stated as:

"Assume that 30% of students in a university take public transportation daily to commute to their college. Suppose 10 of the students are randomly selected.

What is the appropriate probability distribution? Also, what is the probability that at least 2 out of 10 students take public transportation daily?"

I looked into jbstatistics for the answer, but I would just like to cross verify. Kindly tell me which distribution you would choose (poisson/binomial) and also, why?

Thanks,

Chris

• I have answered to the distribution question. If "what is the probability that at least 2..." still needs an answer please consider adding a self-study tag: stats.stackexchange.com/tags/self-study/info and post your own thoughts on how to solve that. Commented Nov 26, 2019 at 8:37
• Hint: every Poisson distribution assigns nonzero probability to the event that 11 or more students out of your group take public transportation.
– whuber
Commented May 19 at 18:46

It is a binomial question as the number of students investigated is fixed at exactly 10 and the probability is "per student".

"Given n = 10, each with a constant probability (p = 0.30) what is the probability of at least 2 positives" is a standard question for a binomial distribution.

Poisson distributions do not have an upper limit.

distr <- dbinom(x=0:10, size=10, p=.3)
plot(0:10, distr, main = "Binomial with n = 10, p = .3", type="h",
xlab="Students out of 10", ylab = "Probability")
text(0:10, y = distr, labels = round(distr,3))

• Okay, interesting. So you're saying that it's a binomial distribution as: 1. Number of examinations are "fixed". 2. Probability is dealt per examination. Can you provide any materials for Binomial and Poisson distributions and their respective practice questions? Commented Nov 26, 2019 at 8:43
• And what do you exactly mean by "Poisson distributions do not have an upper limit"? Any slides/videos/pdfs etc. ok this topic would be thoroughly appreciated. Thanks! Commented Nov 26, 2019 at 8:45
• I could do a google/ecosia search on questions on discrete distributions but that is not really the point of this forum. If you sit at the telephone of a school office and usually there is one phone call per minute, what is the maximum number of phone calls that could appear within an hour? Well, there is no upper limit of phone calls. Could be thousands, even if that is very, very unlikely. So the phone calls have no upper limit, the number of students out of n can never exceed n. That is an upper limit. Commented Nov 26, 2019 at 8:50
• Okay, I think so I get it. I will defn. go online and search for more! And I appreciate seeing a fellow Ecosia user! Thanks for the help! Commented Nov 27, 2019 at 7:58

Further to what @Bernhard has noted, you are after the binomial distribution as you have yes/no trials (did they take public transport or not).

• The experiment results in outcomes that can be classified as successes or failures.
• The average number of successes (μ) that occurs in a specified region is known.
• The probability that a success will occur is proportional to the size of the region.
• The probability that a success will occur in an extremely small region is virtually zero.

A way to show the 'no upper limit' might be goals in football (or any sport where the highest score is not limited (e.g. not tennis)). There could be between 0 and any number of goals in a game. You would have an average number of goals in a game though and from that you can calculate the probability of (for example) six goals in a game. Example here.

• Thanks for the help! I really appreciate it. Especially the example. Have a nice day! Commented Nov 27, 2019 at 7:58

## A simple reason to use the binomial distribution:

You have to use the binomial distribution as other answers suggest but the reason is simpler. You know the probability of one student (30%) and the number of students (10). These are the parameters of the binomial distribution needed to calculate the probability you are after.

Poisson is just an approximation to the binomial distribution. It is invalid in your case. To be valid the probability of one student must be low and the number of students large. Practical numbers are 5% and 20 respectively, clearly not your case.

• "These are the parameters" is correct but it is not a good procedure for deciding which distribution (if either) would apply, because many well-formulated exercises of this sort will include extraneous information.
– whuber
Commented May 19 at 18:47
• This problem has to be solved with a binomial distribution. Using Poisson is a mistake for the reasons provided in the answer. The comment about extraneous information is very vague, please post a concrete a example in another question. Commented May 19 at 23:32
• I am not contesting the answer: it's Binomial. I am pointing out that the arguments you are supplying to support that answer are not valid. No concrete example is going to help demonstrate that.
– whuber
Commented May 20 at 2:35