So I am a bit confused.
Say $1$ person in $1000$ forget to clean their hands after going to washroom. If $10,000$ go to the washroom, what is the probability of $6$ people forgetting to wash their hands?
So, from my knowledge, I would assume that $\lambda$ is the following:
$$\begin{align}
1/1000 &= x/10000 \\
x&=10 \\
\implies& 10/1000
\end{align}$$
Hence, $\lambda$ is $10/10000$, however according to my friend it is $10$ and not $10/10000$. Can someone explain to me which one is correct? I only want to know which $\lambda$ is correct, since I already have a good idea how to solve it once I get this darned lambda thing down.
Best Answer
Your friend is correct. $\lambda$ is the expected number of successes. So if there are $10,000$ trials each with probability $1/1000$, the expected number is $10$ and that is $\lambda$ The idea is that $\lambda$ will change with more trials- if you had $100,000$ people who visited the washroom you would expect $100$ to forget. That is the $\lambda$ Now the variance is lower.
Incidentally, $1/1000$ is vastly low, but that is not a mathematical problem.