When to Add, When to Multiply?

Date: 10/30/2001 at 16:23:25
From: Aliza
Subject: Probability

I am studying probability at high school. When do you add and when 
multiply to compute probability?

Date: 10/31/2001 at 06:59:39
From: Doctor Mitteldorf
Subject: Re: Probability

Dear Aliza,

You're looking for rules that will get you through.  t's not going to
work. In probability more than in any other field, you have to 
understand the situation, and have a feel for it. Doing lots of 
problems will help, even if you feel insecure trying to work them at 
first. It helps especially if you do your best to think about the 
problems on your own. Try this and that - see what makes sense to you.  
The calculation is the easy part, and knowing what to add or when to 
multiply is what it's all about.

Remember that probabilities are all numbers less than 1. When you
multiply them together, they get smaller. When you multiply several
together, they can rapidly get a lot smaller. Now think about unlikely
events: If it's unlikely that one such event occurs, it's "doubly" 
unlikely that two will occur. You multiply probabilities together to 
compute the unlikeliness of this coincidence - two unlikely events 
both occurring together.

Probabilities are like any other positive number; when you add them
together, they get larger. Adding the probabilities for two unlikely 
events gives you a larger probability that either one or the other 
will occur.

When you're looking for the probability that two events, A and
B, will BOTH occur, the probability of this coincidence is small, and 
you multiply the separate probabilities of A and B to get a smaller 
number. When you don't care which happens - either A or B - you can 
add the probabilities to find the separate probability that one or the 
other will happen.

So now I've done it. First I told you that there are no rules for
telling you when to multiply and when to add; then I gave you a rule 
that tells you when to multiply and when to add. 

But now that I've given you the rule, I should also tell you about the
exception. This is the case that really makes the subject hard to 
think about.  

Think about whether it's logically possible for both A and B to occur. 
For example, two people have bought raffle tickets. If A wins the 
grand prize, then B cannot win, and vice versa. In this case, it makes 
perfect sense to add the probability of A winning to the probability 
of B winning, and the result is the probability that "either one of 
them" will win. If there are 1,000 raffle tickets, then A has a 1/1000 
chance of winning and B has a 1/1000 chance of winning. Together, 
their chance of winning is exactly 2/1000.

On the other hand, think about a situation where A or B or BOTH are 
all possibilities. What is the probability that A or B has a birthday 
on Halloween? Let's assume the probability that A has a birthday on 
Halloween is 1/365, and that B has a birthday on Halloween is 1/365.  
Your first thought is that the probability of either A or B having a 
birthday on Halloween is 2/365. This seems right until you ask what 
the probability is for 3 people. Is it 3/365? For 100 people is it 
100/365? For 1000 people, what is the probability that one or more of 
them has a birthday on Halloween? Is it 1000/365? No - that doesn't 
make sense, because 1000/365 is a number greater than one. 

The problem with adding 1/365 + 1/365 = 2/365 is that we've DOUBLE
COUNTED the probability that both A and B have their birthdays on 
Halloween. We can correct the answer by subtracting that tiny 
coincidental probability:

   the right answer is 1/365 + 1/365 - 1/(365*365).

How would you go on to find the probability that either A or B or C 
has a birthday on Halloween? Now what have you double-counted? This is 
harder to think about, because you've double-counted (A and B), but 
you've also double-counted (B and C) and (A and C). And what about the 
REALLY unlikely possibility that A and B and C all have birthdays on 
Halloween? Have you double-double counted it?  Or have you double 
counted the double counting, in which case a minus times a minus is a 
plus, and you need to add it back?

This is confusing, I know, and there are only 3 people. We need a
better way to think about it, especially if we're going to be able to 
handle the question about 1000 people.  

Here's a trick that will solve the problem more easily - but the
drawback is that it takes us back to the situation where adding and
multiplying can become confused, and you can be left wondering whether 
to add or to multiply.  

Think about the probability that A does NOT have a birthday on
Halloween. That's 364/365. Same for B. If you want the probability of 
the "coincidence" that they both don't have birthdays on Halloween, 
you can get that probability by multiplying 364/365 * 364/365. This is 
exactly the right answer.  

The situation that either one or the other or both has a birthday on
Halloween is exactly the opposite of this, so you can subtract from 
one

    1 - 364/365 * 364/365

and get exactly the right answer.

So here it is: the situation where your first thought was that adding
1/365 and 1/365 should work, but actually you needed to multiply, not 
to add. And to make matters more confusing, the difference between the 
two answers is really tiny, because 1 - 364/365 * 364/365 is almost 
the same as 2/365.

I'll finish with a puzzle for you. I've given you two ways to get the
right answer: the other one was 2/365 - 1/(365*365). Which is the 
right right answer?  

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