Two Options in Lottery - Probability

Question

I was thinking about the lottery and came to this question, which is way above my understanding of probability.

Let's say there are 100,000 tickets and 10 IPhone prizes. You can buy as many tickets as you want but can only win one IPhone. (One prize per person.) Let's assume that you bought 10 tickets.

What would be more advantageous: to submit all of the 10 tickets by yourself, or to tell 10 of your relatives to submit the tickets on their own? (Of course we assume that the relatives give the prize to you and don't run away with it!)

Is there a scenario where the other option becomes more advantageous?

Thanks in advance,

Best regards.

Answer 1

Suppose $n$ tickets are winning tickets. If you were to submit all of the tickets yourself, then you would win:

• $1$ iPhone, if $n>0$; or
• Nothing, if $n=0$

Now suppose your family members each submit the tickets instead, and if they win, give their prize to you. In that case, you would always win $n$ iPhones. Specifically:

• $n$ iPhones, if $n>0$; or
• Nothing, if $n=0$

Now, it's important to compare cases: in the case that no ticket wins, then you are left in the same position regardless of which option you picked. This is the same as if one ticket is the winner. In all other cases, choosing your relatives to submit the tickets is more advantageous. Thus, in all cases, it is either equivalent or more advantageous to choose option 2.

Answer 2

Your probability of winning an iPhone in either scenario depends on how many other participants there are in the lottery, and the distribution of tickets amongst them. If all the other $99,990$ tickets are are held by no more than $9$ participants, for instance, then you're certain to win an iPhone in either scenario.

Regardless of the distribution of tickets, however, the probability that none of your $10$ tickets wins an iPhone when you submit them all yourself is exactly the same as the probability that none of them wins when you get $10$ of your friends to submit them. Therefore, the probability that you win an iPhone (i.e. that at least one of your tickets wins) is also the same (as lulu has already pointed out in a comment).