Associated Topics || Dr. Math Home || Search Dr. Math

### 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 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

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

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

- Doctor Mitteldorf, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Probability

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search