# What are my chances of winning?

Let's assume a hypothetical contest. You can join the contest by submitting a ticket, but multiple tickets can also be submitted by the same person. (For the purpose of this question, let's say there are 5000 tickets entered and 300 applicants). Twenty applicants can win, no person can be selected twice. What are the chances of one person winning with n tickets submitted?

• if there is only 1 winner then the chances of a person with n tickets winning is n/5000. Or, are you asking "what is the chance that the person who wins will have n tickets?" – Jacob Jul 24 '14 at 19:06
• Sorry, I forgot to add the crucial part of this question: What if there can be 20 winners? What are the chances of a person with n tickets winning then? – GreySwordz Jul 24 '14 at 19:12
• Suppose there are only three entrants and two prizes. Consider these extreme cases: (1) one person enters $5000-1-n$ tickets, one person enters $1$ ticket, and you enter $n$ tickets. (2) Two people enter $(5000-n)/2$ tickets each and you enter $n$ tickets. If $5000/3 \gg n\gg 1$ then in case (1) you are almost sure to get a prize but in case (2) you are likely to lose. The answer therefore depends on how many tickets each entrant has in the lottery. – whuber Jul 24 '14 at 19:28
• Can the same person win more than one prize? are ticats replaced after drawing? – Jacob Jul 24 '14 at 19:30
• @Jacob Since nobody is allowed to win twice, it makes no difference whether tickets are replaced or not. – whuber Jul 24 '14 at 19:34

Notice that $P(\textrm{Win})=P(\textrm{at least 1 of his tickets are picked})=$
$1-P(\textrm{none of his tickets are picked})$
Now this probability is a little bit tricky to calculate because you need to know proportion of tickets each person has in the contest because what basically happens is lets say that there are $N$ tickets total and you have $n$ tickets the probability you wont win first prize is merely $\frac{N-n}{N}$ but after a one ticket is chosen that isnt you from a person lets say has $m$ tickets than probability you dont win second prize in this situation would be $\frac{(N-n)-m}{N}$. So it changes depending on who is chosen, if a person with a huge protion of the total tickets is chosen first you have pretty good chances of winning a prize there after, versus a bunch of people that have only like one ticket chosen before you.