Odds in Large Tournament vs Smaller Tournament

I have been discussing this issue with a colleague and I invite opinions on this topic.

Scenario:

A) You have one tournament with 10000 entrants.
B) You have another tournament with 1000 entrants.

Payout: Top 10% of the entrants.

Given:

1. All of the blinds are the same.
2. The chip starting stacks are the same.
3. The player skill is not a factor.
4. Every other factor is equal.

My position: You have the same odds of cashing in both tournaments.

My friend's position: Cashing in Tournament A would be significantly harder than cashing in Tournament B. His reasoning is that you have to beat 9000 people instead of 900 people and that you have to dodge more bad beats.

My counter argument: I would also be inflicting bad beats and bad beats are not a static amount of chips.

Please reply with any math proof or statistical analysis for either position.

Thank you!

Will anyone be checking for counterfeit chips ?

10% of the people cash. If everyone's chances are the same then everyone's chances are 1/10.

If everyone's skill is the same, as your question assumed, then it's obvious that in either case, the chance of cashing is 10%. Because that is by definition.

In real life, it depends on skill. If you're below-average in skill, it's easier to cash in the smaller tournament. If you're above-average in skill, it's easier to cash in the larger tournament.

In both, assuming 10 person tables, you have to knock 9 other people out to get to the 10% where you will cash. But that assumes everyone else will have to knock out 9 other people by the time you've done that. It seems more likely you'll have to work harder (knock out many more than 9) before the field gets to 10 % of its start in the larger group. How do tournaments deal with that, anyway? If you have a slow group and/or dealer, people at your table might stay in the running longer if other tables are moving faster (more hph = more chances to bust out). Do they make the survivor at each table wait until all survivors have been determined? Seems like they keep you moving and keep combining down tables.

Quote: beachbumbabs

In both, assuming 10 person tables, you have to knock 9 other people out to get to the 10% where you will cash. But that assumes everyone else will have to knock out 9 other people by the time you've done that. It seems more likely you'll have to work harder (knock out many more than 9) before the field gets to 10 % of its start in the larger group. How do tournaments deal with that, anyway? If you have a slow group and/or dealer, people at your table might stay in the running longer if other tables are moving faster (more hph = more chances to bust out). Do they make the survivor at each table wait until all survivors have been determined? Seems like they keep you moving and keep combining down tables.

There are different types of tournaments. In shootouts, the winner of every table moves on. That's kind of rare, though. Usually they move people around to keep the tables more-or-less balanced.

Anyway, this is making things more confusing than they need to be. The question was about your chances of cashing. The statement was that everyone has the same chance. If 10% cash, and everyone has the same chance, then everyone has to have a 1/10 chance of cashing or the numbers don't add up. You are basically just picking 10% of the entrants at random. Everyone has a 10% chance of being selected regardless of the initial size of the pool.

Your odds are the same.
People will stumble to the finish line, there's no disqual for being too lucky or just surviving until the bubble's burst.
With respect to getting bad beats, if you go all in too often you will eventually lose.
You should be more concerned with blind structure. The more turbo it is, the more you can fall behind and find yourself being forced to go all in with marginal starting hands.
Therefore, you...individually, may have to worry if your playing style is too relaxed for the blinds increases. But the hypothetical you? Same thing.

Thank you for all of your answers!!