There are \(n\) lottery tickets in sale. \(m\) of them are the winning tickets. You have bought \(r\) tickets. What is the probability for you to win in the lottery. P.S. You win if you have one or more winning tickets.

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The probability that a ticket wins is: \[\frac{m}{n}\] The probability that it loses is thus: \[1 - \frac{m}{n}\] The probability that you have \(r\) loosing tickets is thus: \[\left(1 - \frac{m}{n}\right)^r\] The probability that NOT all tickets loose (at least one tickets wins) is thus: \[1 - \left(1 - \frac{m}{n}\right)^r\]

If I have bought more than \(n-m\) tickets, so \(r>n-m\) I will 100% win. But your probability is not equal to 1 even if \(r=n\). Think again.

Right, so we have to take into account the fact that with each loss, the probability of a ticket failing decreases. So the probability that a ticket loses at the start is \[1-\frac{m}{n}\] and for the second one it is: \[1-\frac{m}{n-1}\] So we get \[1 - \prod_{k=0}^{r-1} \left(1 - \frac{m}{n-k}\right)\] Umm how about that?

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