A lottery sells $10,000,000$ tickets with probability of winning at $\large\left(1\over 5000000\right)$
Let $t$ to represent the number of tickets sold. Using the formula $1-\left(4999999\over 5000000\right)\large^t $, the probability of there being a winner is approximately $86.5\%$ (or for no winner, $13.5\%$). Okay, I understand that.
Using the same numbers, what if the lottery guarantees there will be $\large2$ winners.
This is problematic because the probability of $0$ winners calculates as $13.5\%$, which is impossible since we know that there will be $2$ winners out of $\large t$ tickets sold. Ultimately my question and confusion is how would you calculate the probability of winner/no winner in such a scenario?
Let me clarify what I'm basically asking. I know the events are not independent because the lottery is guaranteeing exactly 2 winners out of 10 million. So, how do you calculate probabilities of $X$ winners at various points of tickets sold. For example, they have sold 9,000,000 out of the possible 10,000,000 tickets. What is the probability of only one winner so far? The winners are not selected after 10,000,000; they are picked randomly.
I assume for 1 winner it would be $9,000,000 \over 10,000,000$
But what about non-obvious figures, i.e. 2 winners?
Would it simply be $\left(9,000,000\over 10,000,000\right)\large^2 $