Hi everyone! I was driving back to Chicago from the Twin Cities with my uncle last night and was listening to the NCAA Tourney on the radio. He and I had each bought a tournament square at our local watering hole; for $25, I was able to select a square on a grid of 100. The barkeep pulled numbers out of a hat, and then assigned 0-10 on both the x & y axis of the grid. You won when the last number of the winning team’s score and the last number of the losing team’s score matched up, i.e., Syracuse beat Texas 95-84, so the person whose square corresponded to W5-L4 won that game. As there are 64 games in the tournament, you have 64 “drawings” in which to win your square, and your square can win as many times as possible; it does not get removed from the pool after you win once. Also, you can buy as many squares as you wish (some guys bought 8 squares, however, at $20 a pop, we opted to each buy a single square).
Here’s where things get difficult. We each bought a single square. He believes that his odds of having his square winning are better at the onset of the tournament, where he has 64 games where his square could match the Winners-Losers score, then at the end of the tournament when there are 2 or 3 remaining games. He belives that as there a fewer games, his odds of winning decrease because there are fewer chances. I believe that his odds remain the same through out the tournament, 1 out of 100, and the fact that there are 64 games has no impact on his chances of winning at least once. They neither increase nor decrease!
To put it short, for any particular game, you were right, you have the same odds. However, each game represents another 1/100 chance, so the more games that are being played, the better your chances of winning at least one of them. I hope this is the answer you were looking for.
Statistically speaking, your odds are 1/100 for each game, assuming each outcome is equally possible. These odds add up though, so for two games you have a 2/100 chance to win one, and for the 32 games in the first round your overall chances to win once are 32/100. Your uncle’s right, in other words.
Not quite. If there were 100 games, you wouldn’t be guaranteed to win exactly once, which your formula would imply. Your chances of winning at least one game in the first 32 is actually only about 27.5%. On the other hand, you also have a chance to win more than one game, such that your average number of games won in the first 32 games is 32/100.
Similarly: If you’re getting close to the end of the tournament, and you haven’t won any yet, then you’re in worse shape than if it were the beginning of the tournament and you hadn’t won any. But it’s less likely, at the end of the tourney, that you haven’t won any. And if it’s the end and you have won one or more, then you’re in better shape than you were at the beginning.
How can that be, your odds get better? Here’s what I don’t agree with: There are 64 drawings, during which you have a shot of 1 of 100 to win. So does everyone else. This would mean that there are a total of 6400 possible winning squares, to which you have staked your claim to 100 of them.
An easy way to think about this might be to pretend that you’re betting on a series of coin tosses instead of a series of basketball games. Imagine that someone is going to flip a coin 64 times and you’re asked to wager whether at least one of the tosses will come up heads. Your odds are really good that at least once it’s going to be heads out of 64 tosses: Not quite 100%, because it is entirely possible that it could just happen to come up tails 64 times in a row; but approaching 100% because you have so many tries (sorry, I don’t remember how to do the math exactly). Now imagine you’ve tossed the coin 63 times already, and you’re down to the last toss. What are your chances that you’ll get heads on that last toss? Exactly 50%, just like any coin toss.
That is, at the beginning, you had a really good chance of winning at least one of the upcoming coin tosses. Now, you only have one shot left.
I think the difference between what your father and you are saying boils down to this: Your odds of winning any single coin toss are always 1/2, and your odds of winning the grid game on any single matchup are 1/100. But your chance of winning at least once more in the remaining games goes down as the number of remaining games drops.