Need Help Understanding a Basic Probability Principle

[Saracen Rue]

I don't really know how to explain it right off the bat, so I'm going to use an example first:

Say 2 coins are tossed. Let H1 = a head is tossed first and T2 = a tail is tossed second.
If you create an event space, you get {H1H2, H1T2, T1T2, T1H2}.
Therefore, the Pr(H1) must equal 1/2 and Pr(T2) = 1/2

Next, I want to attempt to find Pr(H1∩T2).
First, I need to find Pr(H1∪T2), which I'll do by using the formula Pr(A∪B) = n(A∪B)/n(universal set):

Pr(A∪B) = n(A∪B)/n(universal set)
Pr(A∪B) = 2+2/4
Pr(A∪B) = 4/4 = 1

However, there is a problem here as I can see by looking at the event space that this can't be true, as there's the event T1H2. So, my first question is why doesn't this work?

So, instead, I can find Pr(H1∩T2) just by looking at the even space, which ends up being 1/4. However, I don't know how I would find this without the event space. So, my second question is how would I find Pr(H1∩T2) by only using formulas (not looking at the even space).

Any help with this would be must appreciated as I honestly can't seem to understand probability at all.

 

[phinds]

How you arrived at your value of 1 for the sequence (H1 OR T2) I don't understand at all and clearly you recognize that your method gives an impossible answer.

As for the sequence H1 AND T2, you get the probability but just multiplying the probability of H1 by the probability of T2 which is 1/2 * 1/2, giving the 1/4 which you got by looking at the solution space.

EDIT: by the way, to get "H1 OR T2" what I do is find 1 - (NOT H1) AND (NOT T2). That is, for "H1 OR T2" to be true, you can't have both H1 being false AND H2 being false so you find that probability, which is easy, and then subtract it from 1 because you're finding the "remainder" probability of what you are really looking for. So I get 1 - 1/2*1/2 = 1 - 1/4 = 3/4 which makes sense. This is a lot faster to DO than to explain.

 

[PeroK]

[QUOTE="Saracen Rue, post: 5095282, member: 521193"

Any help with this would be must appreciated as I honestly can't seem to understand probability at all.[/QUOTE]

With all maths and physics and especially probability, I think it's a bad idea to try to rely on formulas. It's much better if you can understand a problem and then pick a formula to help you solve it. Techniques like simple counting, Venn diagrams and probability tree diagrams are much better tools to help you understand what's going on. Once you understand a problem, you can encapsulate that in a formula. But, I wouldn't try to do it the other way round.

Get a pack of cards and, without using formulas, start looking at some simple questions. E.g. if you have 5 cards, what is the probability that exactly 2 cards are red. Deal a few hands and try to see what's happening. How often do certain patterns emerge? (To begin with, you can assume that each cartd is 50-50 red/black.)

The good thing about probability is that it's out there all around and, unlike most maths, you can experiment with it.

To take the example you have here:

Probability H1 and T2 = (1/2)(1/2) = 1/4. I don't see there's any more to it that.

 

[FactChecker]

Pr(A∪B) = n(A∪B)/n(universal set)
Pr(A∪B) = 2+2/4

You are counting some things twice here. You have to subtract the intersection of A and B to correct for that. But calculating that intersection was the original problem, so this is not the most direct way to solve the problem.