Lotto probability , at least one person independent Given that  N=25.000.000 people played lotto 
 To win:
They have to choose the correct 5 numbers from a list of 45 total numbers 
( You can only choose each number once)
And 1 correct number from a list of 20 numbers
So the chance ( 1 person) wins the lottery is ~0.000004%
So 4 out of 100 millions.
But what is the chance that at least one person wins the lottery out of 25 million players given that they might even choose the same numbers with another player. 
 A: The probability that a given ticket wins is $$p={1\over20{48\choose5}}={1\over34,246,080}$$  The distribution of the number of winning tickets is a binomial distribution, and with such a small probability of winning and so many tickets, the distribution is well approximated by a Poisson distribution with parameter $$\lambda = 25,000,000p$$  The probability of exactly $k$ winning tickets is approximately $$e^{-\lambda}{\lambda^k\over k!}$$
The probability that there is exactly one winning ticket is approximately  $$0.3517949$$
We could also compute this directly from the binomial distribution as $$25000000p(1-p)^{24999999}\approx 0.3517949$$ but the Poisson distribution is more convenient if one starts asking more complicated questions.
A: Let $p$ be the probability that a given ticket wins.
Then the ticket has probability $1-p$ of not winning. If 25 million people choose numbers independently then the probability no one wins is $(1-p)^{25000000}$ and the probability at least one person wins is $1-(1-p)^{25000000}.$
