Conditional Probability

[danhodge]

Whelp, I have no idea where to put this, and even less of an idea whether anyone here can help me, so here goes nothing:

Studying for my Advanced AI module, and I thought "Hey, let's do some of the questions from the textbook! They can't be much harder...". I then proceeded to cry myself to sleep.

Basically, this question asks:

And this is the answer they give:

All I don't understand is, where does the ' A ^ ' go, on the 3rd step? It just disappears and I can't think of any logical reason why

Any help will be forever loved and appreciated

 

[bpm55]

There is nothing here, but the question. The links or images you put in are not present.

 

[danhodge]

There is nothing here, but the question. The links or images you put in are not present.

That's odd, it works fine for me on my desktop and on my phone, can anyone else see it?

 

[carnageX]

I can see them just fine.

That said, I have no idea. Been a while since I've taken statistics or Logic, Sets, & Proofs .

 

[kmanmx]

From what I remember Soulphire is alright at maths, he might be able to help.

Other than that you might want to try https://math.stackexchange.com/ or the math subreddit https://www.reddit.com/r/math/

 

[danhodge]

From what I remember Soulphire is alright at maths, he might be able to help.

Other than that you might want to try https://math.stackexchange.com/ or the math subreddit https://www.reddit.com/r/math/

I thought it might be a long shot, worth asking though, some clever people here 😂

 

[bpm55]

Good news is I can see the question. Bad news I have forgotten most of that math class. Sorry hopefully someone else can help out.

 

[S0ULphIRE]

They just performed a simplification, the probability set of A in union with (B in union with A) is the same no matter how many times you say "in union with A", i.e. you can simplify it down to just B union A. That's why they say that bit about "and the fact that A union A == A"

 

[danhodge]

They just performed a simplification, the probability set of A in union with (B in union with A) is the same no matter how many times you say "in union with A"

But isn't the probability of A ^ B the probability that both will happen at the same time? So wouldn't it be the probability of A happening on its own, plus the probability of both happening at the same time?

Thanks for helping btw, my lecturer for this module has no idea

 

[S0ULphIRE]

But isn't the probability of A ^ B the probability that both will happen at the same time

Yep! Try making two overlapping circles, label A and B. Label the middle overlapping bit A^B

What is the probability that some random point I choose will be in set A^B (the tiny oval in the middle) AND in set A?
Well, set A^B is ENTIRELY within set A. So if it's in set A^B at all, it will 100% be in set A as well. We don't need to calculate that.

Or to state it more clearly, if we have P ( A ^ ( A ^ B)) then the associative law says we can also rewrite that as P((A ^ A) ^ B)
P(A ^ A) = P(A)


Also nice! Which AI course are you doing?