WHY Do We Add or Multiply in Probability?

Last time, we discussed how you know whether to add or multiply (or something else) in compound probability problems (like finding the probability that you will flip heads and roll an even number). But as I’ve said before, it’s often easier to remember a formula if you know why it is what it is. I’ll focus here on two such questions.

Why do we add to find P(A or B)?

Here is a question from 1999:

Why Add Probabilities in an OR Statement?

Can you please answer my question? Why is it that whenever we want to find the probability of, for example, rolling a 2 or a 3 on a die we add up the number of twos and threes? Or when we want to find the probability of getting an ace or king we add up aces and kings? 

It is the OR that I am not getting. Why do we add? 

I have also heard of P(A or B) = P(A)+P(B). I understand when it is one specific thing like picking an ace or rolling an even number, etc., but not when there is an "or." Please try to give me an explanation. Thanks!

Mike is probably very new to probability, as the first question could have been answered directly in terms of the definition of probability: the total number of possible successes (rolls that are either 2 or 3), over the total number of possible outcomes. The very word “or” often causes confusion in math, because its application can feel backward. But this was early in my time at Ask Dr. Math, and I missed that issue (“Why add the 2’s and 3’s, when it says ‘2 or 3’ ?”). I may have answered at a higher level than I should have. But it makes a better answer to the bigger question this way, so I’ll use it here, and discuss the “or” issue below.

I mentioned the definition, and then illustrated the general idea in terms of areas (which I could just as well have described as Venn diagrams, representing sets):

The probability of an event is the ratio of the number of (equally likely) cases in which the event will happen, to the total number of possible cases. 

You can think of it in terms of area: if I draw a circle on the floor and drop an object in such a way that it is equally likely to fall at any point on the floor, the probability of its falling in that circle is the ratio of the area of the circle to the area of the floor:

    +-------------------------------------------+
    |                                           |
    |                                           |
    |                                           |
    |                                           |
    |                                           |
    |                                           |
    |                                           |
    |           *****                           |
    |        ***.....***                        |
    |       *...........*                       |
    |      *......A......*                      |
    |      *.............*                      |
    |       *...........*                       |
    |        ***.....***                        |
    |           *****                           |
    |                                           |
    +-------------------------------------------+

Now if I draw two circles that don't overlap, then the probability of its falling in circle A OR circle B is the ratio of the area of both circles together to the area of the floor:

    +-------------------------------------------+
    |                          *********        |
    |                       ***---------***     |
    |                     **---------------**   |
    |                    *-------------------*  |
    |                   *---------------------* |
    |                   *----------B----------* |
    |                   *---------------------* |
    |           *****    *-------------------*  |
    |        ***.....***  **---------------**   |
    |       *...........*   ***---------***     |
    |      *......A......*     *********        |
    |      *.............*                      |
    |       *...........*                       |
    |        ***.....***                        |
    |           *****                           |
    |                                           |
    +-------------------------------------------+

But the total area of A and B is just the sum of the areas of the two circles, so the probability of its landing in either A or B is the sum of the two probabilities. This is what we are doing whenever we find the probability of either of two mutually exclusive events (that is, both A and B can't happen at the same time), such as getting a King OR a Queen.

So the numerator of the probability is the total number of elements in the two sets – we add them. And when you add the numerators of two fractions with the same denominator, you are adding the fractions.

My picture assumes that the two events don’t overlap – that they are mutually exclusive. This was assumed in Mike’s question, and leads to the special-case formula, P(A or B) = P(A) + P(B).

But what if the events overlap – if there are outcomes that are part of both events?

Now if A and B overlap, then we won't be able to tell what the probability of either event happening will be unless we have additional information to tell us how much they overlap:

    +-------------------------------------------+
    |                                           |
    |                                           |
    |                                           |
    |                     *********             |
    |                  ***---------***          |
    |                **---------------**        |
    |               *-------------------*       |
    |           *****--------------------*      |
    |        ***...*x***------B----------*      |
    |       *......*xxxx*----------------*      |
    |      *......A.*xxxx*--------------*       |
    |      *.........**xx*------------**        |
    |       *..........***---------***          |
    |        ***.....***  *********             |
    |           *****                           |
    |                                           |
    +-------------------------------------------+

So if you want to find the probability of getting, say, a King OR a red card, you have to know something more, namely that these two events are independent, so that the probability of A AND B is P(A) * P(B). But that's a different matter.

Here the picture only hints at the general formula. In a case like this, we can count the outcomes by adding A and B, and then subtracting the overlap (intersection), which is counted twice when we add. This intersection is A and B; so the general formula is P(A or B) = P(A) + P(B) – P(A and B). Rather than explain this (which seemed likely to be beyond what Mike had learned yet), I just referred to Inclusive Probabilities, where Dr. Anthony had said more or less what I just said, with an example that you might like to read.

But how does what you just did mean “or”?

I mentioned above that “or” can cause confusion. Here is an answer I gave about this in 2008, not in the context of probability, but in an elementary counting question:

The Difference between And and Or

My son had a question that was marked wrong on his paper.  He pointed out to me that by the way it was worded, he felt as though he were correct.  Here is the question:  There are 3 knives, 4 spoons, 4 forks.  What fraction of the utensils are spoons OR forks?

He answered 4/11 and was told the teachers edition says 8/11.  I understand the way he read it to be OR meaning one or the other.  If it's 8/11, shouldn't it be worded spoons AND forks?  If the answer is 8/11, I want my son to understand why.

The question here is, essentially, “Wouldn’t we add when it says and, rather than or, because we are putting two sets together?” In response, I emphasized that the usage in talking about sets (or events) focuses on each individual element of a set, not each set as a whole; the wording can end up seeming reversed if we look at it the wrong way:

When we talk about the set of things that are A AND B, we mean that EACH of those things must be BOTH A and B.  Nothing is both a spoon and a fork!  (At least not in this problem.)  So "and" would have been inappropriate.  There are no utensils that are spoons and forks.

When we talk about the set of things that are A OR B, we mean that EACH of them may be EITHER A or B.  That is, we are including in the set BOTH those that are A, AND those that are B.  This is where the confusion and ambiguity come in!  There are 8 utensils that are spoons or forks.

Your son read it in a way that is commonly used in nontechnical English, taking "How many are A or B" to mean two separate questions combined: "How many are A, how many are B".  I can see how that could be tempting in this case; the two numbers happen to be the same, so he could take the question to mean "How many are A (which is also the same as the number that are B".  If there had been 3 spoons and 4 forks, that interpretation would not have made as much sense; the best answer he could give would be "3, or 4".  We don't combine questions like that in math, to avoid confusion.

Dr. Riz supplemented this with a different example (and a more creative conclusion):

In logic, an AND statement is only true if both parts of it are true. If I say, "I am in Vermont AND I am in New Hampshire" the only way that can be true is if I am standing on the border with one foot in each state.

An OR statement is true if either part is true.  If I say, "I am in Vermont OR I am in New Hampshire" that statement is true as long as I am in either state (it's also true if I'm straddling the border).  The only way an OR statement is not true is if both parts are false, such as if I were standing in Massachusetts when I made my statement about being in Vermont or New Hampshire.

With your question about utensils being spoons OR forks, I count every utensil that is either a spoon or a fork, giving 8 of the 11.  If I were asked what fraction of the utensils were spoons AND forks, there would be zero since the utensil would have to be both things.  There IS a utensil you sometimes see in fast food places which is a spoon shape with teeth on the front edge, and it's generally referred to as a "spork", a combination of spoon and fork.  That's what I'd need for a utensil to be considered a spoon AND a fork.

So in Mike’s probability question, the number of cards that are aces or kings is found by adding the number of aces and the number of kings.

Why do we multiply to find P(A and B)?

This question came from Hannah in 2009:

Why Do We Multiply the Probabilities of Independent Events?

I would like to know the reasoning behind this topic.

P(A and B)= P(A)xP(B)

Why do we have to multiply the two probabilities, why can't we add them together?  Thank you very much.

I replied,

It may be clearer to you if you think of probability as the fraction of the time that something will happen.  If event A happens 1/2 of the time, and event B happens 1/3 of the time, and events A and B are independent, then event B will happen 1/3 of the times that event A happens, right?  And to find 1/3 of 1/2, we multiply.  The probability that events A and B both happen is 1/6.

Note also that adding two probabilities will give a larger number than either of them; but the probability that two events BOTH happen can't be greater than either of the individual events.  So it would make no sense to add probabilities in this situation.

Six years later, Mujari wrote to ask for a further explanation, and I drew some pictures to make it concrete, which we added into this page because it was a helpful clarification. (Thanks, Mujari!) I’ll let you look through it and see how it works for you.