Question

Binomial vs. Poisson Distribution

I have 10 chairs available per table at a charity fundraiser.

Looking now over the whole venue, there are 100 tables with 10 chairs each. I have sold every ticket, but I have used all these ticket sales to pay for the celebrity speaker. Let’s say that there is a probability of 80% that someone who purchased a ticket to the event will show up. For every person that shows up, I will make $100 on average in additional fundraising from auctions, raffles, etc. For every person that does not show up, I will lose $100 in food and hall rental costs. What is the chance that I will make a profit on this fundraiser?

Answer 1

Let's assume X number of people show up out of 100*10=1000 people, where X is random variable that follows Bin(1000,p=0.8) distribution.

You will make a profit if the number of people that show up is greater than the number of people that do not show up, since you make same amount of profit and loss given a person shows up or does not show up.

This can happen when X>500.

Therefore, we need to calculate P[X>500].

But a binomial distribution for large n values can be approximated by a Poisson distribution with parameter \lambda=mean of the binomial distribution in case the mean is finite.

Here, mean=1000*0.8=800

That is it's certain that you will have a profit.