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 
three branches and then add:

(.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%
              good                bad
                          
            98%/    \2%      98%/   \2%
           good      bad     good    bad



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/