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### Probability Tree Diagrams

```
Date: 8/22/96 at 1:20:30
From: Philip Carter
Subject: Probabilities

I wonder if you would help to settle an arguement by providing answers
to the three problems below:

1. A woman has two children. What are the odds that both are boys?

2. Charlie hits the target 80 times in 100 shots.  Jim hits the target
90 times in 100 shots.  What are the chances that the target will be
hit if each fires once?

3. In 1946, statistics showed that 2 percent of fruit boats arrived
with their cargoes ruined. If two boats arrived, what was the
probability that both cargoes were ruined?

Philip Carter
```

```
Date: 8/22/96 at 2:36:22
From: Doctor Paul
Subject: Re: Probabilities

Let's set up a tree diagram:

50% /               \ 50%
boy                 girl

50% /   \50%      50%/   \50%
boy     girl      boy    girl

To find the odds, just multiply across the path that leads to two
boys:

.50 * .50 = .25 or 25 percent.

Had you wanted to know the odds of a boy and a girl, you would have
multiplied across the path that leads to a boy and a girl and then add
that to the path that leads to a girl and then a boy.  They both
satisfy the criteria so you add them.

2. Again, let's set up a tree diagram...assume that Charlie goes
first.

80%/                 \20%
hits                   misses

90%/   \10%      90%/     \10%
hits     misses   hits      misses

Let's see which branches lead to the target being hit:

if it is hit and then hit again, that counts; if it is hit and then
missed, that counts; if it is missed and then hit, that counts; if
both miss it, that doesn't count... so let's multiply along the first

(.8 * .9) + (.8 * .1) + (.2 * .9) = .72 + .08 + .18 = .98,
or 98 percent.  Pretty good odds, eh?

3. Tree diagram again... (are you noticing a theme?)

98%/              \2%

98%/    \2%      98%/   \2%

Both ruined:  .02 * .02 = .0004 = .04 percent (pretty low)

I hope this helps you out.  These problems are easy if you use tree
diagrams!

Regards,

-Doctor Paul,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Probability
Middle School Probability

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