Poisson's distribution for probability

Question

This question is more for learning purposes than anything however I came across this while trying to solving the following problem:

The odds of winning the lottery are 1 to 50000 million. This week, 50 million tickets are sold for the latest drawing. What is the probability that at least one winning ticket was sold?

A friend of mine told me this had something to with Poisson's distribution. What is it exactly about Poisson's distribution that allows you to solve this problem? From my own research the formula for Poisson's distribution is $$P(X=x)=\frac{\lambda^xe^{-\lambda}}{x!}$$ where $x$ is a discrete variable representing the number of occurences of an event and $\lambda$ is the mean number of occurences of the event. Based on this, I'm assuming the 1 in 50000 million is just supposed to represent any arbitrarily small number and is $\lambda$ equal to this probability however I don't really understand the concept well enough to apply it. Can someone explain Poisson's distribution in general and in the context of this problem?

Answer 1

In general

The poisson distribution counts the number of times an event will happen in a given time frame. However, there is a special case of it in the poisson limit theorem in which it can be used to approximate binomial distributions with exceedingly low odds of having two or more events. It is also called the law of rare events for this reason.

In the context of this problem

The odds $1$ to $50,000$ million means (with $a$ as $50,000$ million) that the probability of winning the lottery is $\frac{1}{a}$. We may also assume that each person winning is independent, cause we can have at least 1 winner, and it makes no sense for them to be dependent. Thus we can model this with $X$ equal to the number of winners, $X \thicksim \operatorname{bin}(50,000,000, \frac{1}{a})$ From here, we can approximate the binomial because the power $(1 - \frac{1}{a})^{50000000}$ might give your computer issues or take lots of time. So we would like to check if we can use a normal approximation, or a poison approximation. I assume that you are aware of the central limit theorem and the poisson limit theorem that justify both of these, as your friend mentioned it, so I will skip them. Instead note that, I am approximating here because it’s in my head, I will edit in the morning with exact numbers $$np(1-p) < \frac{1}{1000} \\ np^2 < \frac{1}{1000000}$$ So a normal approximation is far from accurate, while a poisson approximation seems very good. Then, we let $\lambda = np = \frac{1}{1000}$, and let $X \thicksim \operatorname{Poi}(\lambda)$ we can find: $$\mathcal{P}(X \geq 1) = 1 - \mathcal{P}(X=0) = 1 - \frac{\lambda^0}{0!} e^{-\lambda} \approx 0.001$$ Thus, the odds that we have at least one winner are approximately $0.001$.

Conclusion/TLDR

A poisson distribution is used in two cases (mainly):

• A model of the amount of events occurring over a given period of time
• An approximation for the binomial distribution with exceedingly low odds of two successes or more

In this problem, we use it as an approximation for the binomial distribution, because two or more people winning the lottery is exceedingly rare with those odds.