2

In a lottery with 1000 numbers, there is only one prize in each draw. Antonio buys 2 tickets for a single draw and Paulo buys 2 tickets, one for each of 2 draws. Determine which of the two players has the best chance of winning a prize.

My answer:

For me Antonio has a 2/1000 chance of winning. Paulo only has a 1/1000 chance, because the draws are independent and different events, so his chance of winning will always be 1/1000. Am I right?

Marco Oliveira
  • 21
  • 3
    Your calculation for Paulo is wrong. He might win in the first draw or in the second draw (and indeed might win in both draws). – Henry Mar 15 '25 at 19:26
  • Related: https://math.stackexchange.com/questions/122053/is-it-better-to-play-1-on-10-lottery-draws-or-10-on-one-lottery-draw – Henry Mar 15 '25 at 19:28
  • Your answer is wrong. Imagine Marco, who buys only 1 ticket for the 1st draw. You say that Marco and Paulo have same probability to win, it is obvious that is is wrong. I suggest you to answer this other question : Determine which of the 2 players Antonio or Marco has the highest probability to loose his 2 pesos. – Lourrran Mar 15 '25 at 19:29
  • So the probability of Paulo winning is 1/1000 + 1/1000 = 2/1000? – Marco Oliveira Mar 15 '25 at 19:46
  • Hint: compute the probability that Paulo loses both draws and take the complement of that. It will still be less than .002 but not by as much as you think. – John Douma Mar 15 '25 at 19:46
  • @JohnDouma I understand now. I thought that because they are different events, that is, different experiments, the chance of winning should be analyzed per event. – Marco Oliveira Mar 15 '25 at 19:52

1 Answers1

1

You are right that Antonio's odds are better, but you have incorrectly calculated Paulo's.

Imagine that there are only two numbers, so the lottery is like a coin flip - you can buy a ticket on heads or one on tails. If Antonio buys both heads and tails on the first game he wins for sure. Paulo buys heads on each of two games. One fourth of the time he loses both games, one fourth he wins both, half the time he wins once. So he is less sure to win.

If the payoff is $p$ for the winning ticket then Antonio collects $p$ for sure. Paolo collects $2p$ with probability $1/4$ and $p$ with probability $1/2$, so his expected payoff is the same $p$. Unlike Antonio, he might collect more and he might collect less. Of course each is out the cost of two tickets.

You can make similar calculations when there are $1000$ tickets. But if the game is fair and the payoff $p$ is twice the price of the ticket then Antonio

Ethan Bolker
  • 103,433