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

### Probability Trees

```
Date: 07/26/98 at 10:12:22
From: charla anderson
Subject: Probabilities

A machine makes bottle caps and packs them automatically into boxes.
One box in ten has a defective bottle cap.

a) What is the probability that each of the next four boxes will
have a defective bottle cap?

b) What is the probability that two boxes out of the first four will
have a defective bottle cap?

Charla
```

```
Date: 07/26/98 at 17:28:32
From: Doctor Ken
Subject: Re: Probabilities

Hi Charla,

These kinds of probability problems can get pretty tricky, but if you
master them, they can be pretty satisfying.

One way to approach these problems is to use probability trees. Here's
how they work:

Let's say the probability of having a sunny day is 0.7, i.e. there's a
70 percent chance that each day will be sunny (this is a funny weather
system in which one day's weather has no effect on the next day's
weather). I can draw a tree like this:

start
/   \
/     \
0.7/       \0.3
/         \
sunny        cloudy
/\             /\
/  \           /  \
0.7/    \0.3   0.7/    \0.3
/      \       /      \
sunny   cloudy  sunny    cloudy

Now, what does this diagram mean? Well, we can use it as a tool to find
probabilities. For instance, what's the probability of having two sunny
days in a row? We start at the top, choose the sunny path, and choose
the sunny path again:

start
//   \
//     \
0.7//       \0.3
//         \
sunny        cloudy
//\             /\
//  \           /  \
0.7//    \0.3   0.7/    \0.3
//      \       /      \
sunny   cloudy  sunny    cloudy

Now we multiply all the numbers we passed along the way:
0.7 * 0.7 = 0.49. That means there's a 49% chance of ending up at the
sunny-sunny "leaf" on the tree. If you want to, you can do all the

start
/   \
/     \
0.7/       \0.3
/         \
sunny        cloudy
/\             /\
/  \           /  \
0.7/    \0.3   0.7/    \0.3
/      \       /      \
sunny   cloudy  sunny    cloudy
0.49     0.21   0.21      0.09

Now, what's the probability of having a sunny day and then a cloudy
day? Reading the number at the end of the sunny-cloudy path tells us
that it's 0.21. Piece of cake!

So what's the probability of having at least one sunny day in two days?
Well, we have to figure out which "leaves" on our tree satisfy the
condition. Sunny-sunny, sunny-cloudy, and cloudy-sunny do, but cloudy-
cloudy doesn't. So we add up the probabilities of the three leaves:
0.49 + 0.21 + 0.21 = 0.91, which is the answer.

Now let's see if we can tackle your problem. The probability of any
box having a defective bottle cap in it is 0.1 (1 divided by 10). So
boxes, we'll make it four branches long. I'll use g for good and b

start
/     \
/             \
0.9/                     \0.1
/                              \
g                                   b
/    \                             /    \
0.9/          \0.1                0.9/          \0.1
/               \                  /                \
g                b                 g                 b
/  \             /  \              /  \              /  \
0.9/    \0.1     0.9/    \0.1      0.9/    \0.1      0.9/    \0.1
/      \         /      \          /       \         /      \
g        b       g        b        g         b       g        b
.9/\0.1 0.9/\0.1 0.9/\0.1 0.9/\0.1 0.9/\0.1 0.9/\0.1 0.9/\0.1 0.9/\.1
/  \     /  \     /  \     /  \     /  \     /  \     /  \     /  \
g    b   g    b   g    b   g    b   g    b   g    b   g    b   g    b

You can use the same method as in my example: to find the probability
that each of the next four boxes will have a defective cap, follow the
b-b-b-b branch. To find the probability that two of the next four boxes
will have bad caps, look at all the branches with exactly two g's and
two b's. I'll let you figure out which branches those are (there are
six of them). Then add those branches' final numbers together.

Good luck!

- Doctor Ken, The Math Forum
Check out our web site! 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