Let's say I'm simulating the result of a tournament where 4 teams play each other, for a total 6 games. They get, say, 1 point for winning and 0 points for loosing.

I want to get a sense of the end distribution. For example, the probability that team B finishes at least third.

The sample space is finite and is a list of possible wins and losses for each team, or alternatively, the sequence of results of each game. (The sample space can also be taken to be the points for each team, if that makes things easier)

I assume games are independent and construct a matrix of probabilities of outcomes between teams (e.g. team A wins with probability 70% over team B). I then use these probabilities to draw from each game, which gives me a draw from the final distribution.

As the $n$ in the number of draws goes up, I hope that I can approximate the true distribution (and answer questions such as "what's the probability that team C has more than $X$ points").

Three questions

1. Is this a Monte Carlo simulation / a naive way of drawing from a distribution?

2. What is the relevant convergence concept that tells me that as the number of draws goes to infinity, my draws converge in (say) distribution to the true distribution?

3. What are the assumptions needed for convergence? What in this setup would make this simulation not converge?

As always, thanks for your time!

Let's say I'm simulating the result of a tournament where 4 teams play each other, for a total 6 games. They get, say, 1 point for winning and 0 points for loosing.

I want to get a sense of the end distribution. For example, the probability that team B finishes at least third.

The sample space is finite and is a list of possible wins and losses for each team, or alternatively, the sequence of results of each game. (The sample space can also be taken to be the points for each team, if that makes things easier)

I assume games are independent and construct a matrix of probabilities of outcomes between teams (e.g. team A wins with probability 70% over team B). I then use these probabilities to draw (probabilistically) from each game, which ends up giving me a draw from the end distribution (the final results).

As the $n$ in the number of draws goes up, I hope that I can approximate the true distribution (and answer questions such as "what's the probability that team C has more than $X$ points").

Three questions

1. Is this a Monte Carlo simulation / a naive way of drawing from a distribution?

2. What is the relevant convergence concept that tells me that as the number of draws goes to infinity, my draws converge in (say) distribution to the true distribution?

3. What are the assumptions needed for convergence? What in this setup would make this simulation not converge?

As always, thanks for your time!

Let's say I'm simulating the result of a tournament where 4 teams play each other, for a total 6 games. They get, say, 1 point for winning and 0 points for loosing.

I want to get a sense of the end distribution. For example, the probability that, team B finishes at least third.

The sample space is finite and is a list of possible wins and losses for each team, or alternatively, the results of each game. (The sample space can also be taken to be the points for each team, if that makes things easier)

I assume games are independent and build a matrix of probabilities of outcomes between teams (e.g. team A wins with probability 70% over team B). I then use this probabilities to draw from each game, which gives me a draw from the final distribution.

As the $n$ in the number of draws goes up, I hope that I can approximate the true distribution (and answer questions such as "what's the probability that team C has more than $X$ points").

Three questions

1. Is this a Monte Carlo simulation / a naive way of drawing from a distribution?

2. What is the relevant convergence concept that tells me that as the number of draws goes to infinity, my draws converge in (say) distribution to the true distribution?

3. What are the assumptions needed for convergence? What in this setup would make this simulation not converge?

As always, thanks for your time!

Let's say I'm simulating the result of a tournament where 4 teams play each other, for a total 6 games. They get, say, 1 point for winning and 0 points for loosing.

I want to get a sense of the end distribution. For example, the probability that team B finishes at least third.

The sample space is finite and is a list of possible wins and losses for each team, or alternatively, the sequence of results of each game. (The sample space can also be taken to be the points for each team, if that makes things easier)

I assume games are independent and construct a matrix of probabilities of outcomes between teams (e.g. team A wins with probability 70% over team B). I then use these probabilities to draw from each game, which gives me a draw from the final distribution.

As the $n$ in the number of draws goes up, I hope that I can approximate the true distribution (and answer questions such as "what's the probability that team C has more than $X$ points").

Three questions

1. Is this a Monte Carlo simulation / a naive way of drawing from a distribution?

2. What is the relevant convergence concept that tells me that as the number of draws goes to infinity, my draws converge in (say) distribution to the true distribution?

3. What are the assumptions needed for convergence? What in this setup would make this simulation not converge?

As always, thanks for your time!