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Multiplying and adding probabilities
By Sarah Hindhaugh on Thursday, December 05, 2002 - 10:00 pm:

Most of us are all taught at school to multiply along the tree when its an 'and' question and to add if its an 'or'. But, ive been asked to find out why we do so. agggggghhhh. Please help

By William Hall on Friday, December 06, 2002 - 12:33 am:

Sarah,

One of the ways we can build up probability theory is to specify a set of basic rules for probabilities and work up from there.

I'm going to introduce a bit of notation here; if you don't understand what I'm saying just say!

Suppose that we have an experiment etc. that can a number of outcomes O1, O2, etc., with all of these outcomes belonging to a set W. We call this set the sample space. We define an event to be a subset of this sample space. So, for example, in the case of tossing a coin, the sample space would be (Head, Tail), and the event 'get a head' would simply be the set (Head).

We can define a probability function P(A) for events A as follows:
1) 0 <= P(A) <= 1 for all A subsets of W.
2) P(W) = 1 (i.e. the probability of one of the events happening is 1)
3) P(A or B) = P(A) + P(B), given that A and B or disjoint.

Pay particular attention to this third rule. It basically says that, given two events that cannot occur simultaneously (that is what disjoint means), the probability of one or the other happening is simply the sum of the individual probabilities. So, the 'or' rule is in fact occurs by definition of the rules or 'axioms' of probability.

Now for the 'and' rule. Now doubt you have come across the conditional probability formula

P(A and B) = P(A | B)P(B)

(we have just rearranged the formula here). This is how we define conditional probability. Now, if A and B are independent events then we must have P(A | B) = P(A) (as P(A and B)=P(A)P(B), again by definition for independent events). The 'and' rule can only be used only when events are independent; in general we have to use the formula above. No doubt in many of the examples you have done, the events considered are obviously independent (e.g. drawing out balls from a hat with replacement). Again, you can see that the 'and' rule in these special cases comes about through definition.

So why these definitions? The first three definitions come about purely because we can use them to build up a consistent theory of probability; from these basic rules we can build up the rest. In particular, it simply makes sense that if two events cannot occur simultaneously (i.e. they are mutually exclusive) that we add there probabilities.

The 'and' rule is defined as such because it follows from what we see in everyday life. If we toss a fair coin twice, we see that in the long run the probability of getting two heads is 1/4 i.e. 1/2 * 1/2, and so on. Hence we define 'independence' in the way stated above.

However, there are certain obscure systems that do not obey this rule; for example, pairs of particles in a quantum mechanical system do not seem to obey this in certain conditions. However (particularly if you are confused by this statement!) it's best not to worry about this.

In summary, the rules of probability are built so that they fit in with the probabilistic behaviour we see in everyday life.

Hope this helps!

Bill

By Arun Iyer on Friday, December 06, 2002 - 10:03 am:

You can verify these rules by any simple example.
Ex: suppose you are to pick one white ball out of 3 white balls and one red ball out of three red balls..In how many ways can u pick the white ball?In how many ways can u pick the red ball?In how many ways both these work can be done simultaneously?

Now consider the probabilities of the events and u should realise why the probabilities are multiplied when two or more events occur independently.

By William Hall on Friday, December 06, 2002 - 11:20 am:

This is just a combinatorial (counting) way of seeing that the product rule for independent events is a sensible definition - by counting up all the possible ways of an event and dividing by all possible outcomes is one way of establishing a probability; we see for independent events that if we multiply the two probabilities together, or count the number of ways both events can occur and divide it by the total number of outcomes, we get the same answer.

Just a quick point about Arun's example - the probability of both events is clearly 1, but the idea is the same; an equivalent example would be two bags full of white balls and black balls, and being asked to work out the probability of drawing a white ball from each.

Bill