I'm using these pages to learn about Bayes' Theory for the first time, and I have recognized that I am confused. How exactly does "P(X /\ Y)" work?

By the definition given I thought that it meant "The probability that X and Y are true". But that doesn't seem to be how it's being used in the equations that follow. Take Example 1, which has P(red/\round)/P(round)= 3/(3+4). But isn't the probability that an object drawn from the bag is both red and round the number of red and round objects divided by the total number of objects (3/10)? So why is it 3 instead of .3? Or, given that we know the object is round, shouldn't the probability of it being red and round be the number of round red objects divided by the total number of round objects (3/7)? How exactly does 3 end up being the right answer here?

Does P(X/\Y) not actually stand for the probability that X and Y are true but something else? I recognize that there are 3 red and round objects, so that must be where the answer is coming from, but that doesn't seem like a probability to me, just a statement of the number of objects that are both round and red. And now that I look at it, it seems like P(round) is doing the same thing. Is P(round) not meant to be the probability that an object in the bag is round? But if it's just standing in for the total number of round then why isn't it "X/\Y/Y"?

In any case, whether here on on the main introduction page, and explanation of how to calculate P(X/\Y) would be helpful.

It would be nice if there could be a hyperlink to a page explaining what a relative odds function is. Without it the second half of this page is unhelpful to those who don't know what a relative odds function is.

I don't know if this is helpful or not, but, as someone who is genuinely trying to use this to learn Bayes' theorem and doesn't already understand it, I found the following confusing:

When you introduce P(X) you don't explicitly show how those cash out. I eventually figured out the proper way to do it after reading the whole page, but I was a bit confused. Just something simple like "P(sick)=.2". Maybe that seems obvious, but it wasn't until I tried to do the example equations on my own that I realized I wasn't actually sure how "P(X)" translated into numbers in an equation.