Question:
A city's lottery works in the following way: An individual selects 6 numbers from the first 30 numbers. The city then selects 6 numbers from the first 30 numbers. If the individual selects the same 6 numbers as the city selected, then they win the lottery.
A lottery ticket costs 1 dollar, and the lottery winner receives $500,000 if he or she wins. 800,000 people are expected to play the lottery.
What is the probability that the city loses money on the lottery?
Attempt: I know that the city loses money if 2 or more individuals win.
The probability of someone selecting the correct six numbers is $\frac{{6 \choose 6}}{{30 \choose 6}}$
Not sure how to proceed after this.
Hint: Denote with $N$ the number of winners. Then $N$ is a binomial random variable with $n=800000$ and $p=\frac{\dbinom{6}{6}}{\dbinom{30}{6}}$. The probability that the city loses money (is indeed the probability that there are 2 or more winners) is $$P(N\ge 2)=1-P(N \le 1)$$
Because $p$ is very small and $n$ very big, you can also approximate $N$ by a Poisson random variable with $λ=np$ (check first that $np<5$).