Why Add Probabilities in an OR Statement?

Date: 05/25/99 at 21:32:57
From: Mike
Subject: Probability: Why do we add 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!

Date: 05/26/99 at 12:52:43
From: Doctor Peterson
Subject: Re: Probability: Why do we add an OR statement?

Hi, Mike. Good question!

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.

In the examples you give, you can work it out very easily: just count 
the number of cards that are either a King or a Queen, and you'll find 
that you're adding the number of Kings and the number of Queens. 
Divide the sum by the total number of cards, and it's the sum of the 
two probabilities:

                 N(A or B)    N(A) + N(B)   N(A)   N(B)
    P(A or B) = ----------- = ----------- = ---- + ---- = P(A) + P(B)
                N(universe)   N(universe)   N(U)   N(U)

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 is a related answer in our archives:

   Inclusive Probabilities
   http://mathforum.org/library/drmath/view/56608.html   

I hope this helps!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/