Simple question about sets in statistics

[Nikitin]

Let's say you have 100 tickets of type A, and 100 tickets of type B in a box. Let's also say the probability to draw ticket A, for whatever reason, is twice that to draw ticket B.

Is this problem, for all intents and purposes, mathematically equivalent to having 200 type A tickets and 100 type B tickets with the probability of drawing both ticket A and B being equal?

The reason I'm asking is that Bayes' rule and so on seems to be based on the thinking that every single element in the sample space $S$ has an equal probability to be "picked" to any other element in $S$..

 

[Stephen Tashi]

Is this problem, for all intents and purposes, mathematically equivalent to having 200 type A tickets and 100 type B tickets with the probability of drawing both ticket A and B being equal?

I'd say yes, if you only draw one ticket. However, mathematically, my answer can be disputed. The event "The ticket is type A" in the probability space with a total of 200 tickets is technically not the same event as "The ticket is type A" in the probability space where there are 300 total tickets. They are two different events, which have the same probability.

The reason I'm asking is that Bayes' rule and so on seems to be based on the thinking that every single element in the sample space $S$ has an equal probability to be "picked" to any other element in $S$..

It is true that the equation defining conditional probability refers to events in different probability spaces. Suppose the most detailed results that we can describe are called "outcomes" (e.g. the die lands with 5 up). The "events" in the sample space are sets whose elments of outcomes. They may a set of single outcomes (e.g. the die lands with 5 up) or a set of many outcomes (e.g. the die lands with a 3,4 or 5 up). If we consider the events involved in the expressions $P(A|B)$ and $P(A\cap B)$ they are the same as sets of outcomes. However they are events in two different probability spaces. The event whose probability is $P(A|B)$ is in a probability space where the outcomes are only the outcomes in the set $B$. The event whose probability is $P(A \cap B)$ is in a probability space that includes all the outcomes of $A$, all the outcomes of $B$ and possibly other outcomes in addition.