Aircraft Disappearances

Date: 01/29/98 at 21:15:25
From: Jenny Ko
Subject: conditional probability

Seventy percent of the light aircraft that disappear while in flight 
in a certain country are subsequently discovered. Of the aircraft that 
are discovered, 60% have an emergency locator, whereas 90% of the 
aircraft not discovered do not have such a locator.  

Suppose a light aircraft has disappeared.

a. If it has an emergency locator, what is the probability that it   
   will not be discovered?

b. If it does not have an emergency locator, what is the probability 
   that it will be discovered?

Date: 01/29/98 at 21:46:48
From: Doctor Gary
Subject: Re: conditional probability

Do you know what "conditional probability" is? It's just a fancy way 
of saying that we already know something. Here's how it can work:

If I choose an integer at random from 1 to 10 inclusive, the 
probability that it is 6 is one in ten, or .1.

However, if I choose an EVEN integer at random within the same range 
of numbers, the conditional probability that it is 6 becomes one in 
five, or .2, because the condition that the number be even means that 
we don't have to consider some of the possibilities we might otherwise 
have to take into account. 

Perhaps we could understand the information in the story better if we 
imagined that there were 100 light aircraft that disappeared in 
whatever certain country the story is concerned with.

Seventy of them would be recovered, and 60% of THOSE SEVENTY have an 
emergency locator.  60% of 70 is 42, so we can break those aircraft 
that disappeared and were later discovered into two groups:

   42 had emergency locators
   28 didn't have emergency locators.

We're told that 90% of the aircraft not discovered didn't have a 
locator, but we have to use our memory and common sense to notice that 
this is 90% of THIRTY planes, or 27. We can now group the planes that 
weren't discovered:

    3 had emergency locators
   27 didn't have emergency locators

>Suppose a light aircraft has disappeared?
>a. If it has an emergency locator, what is the probability that it 
    will not be discovered?

Of every 100 light aircraft that disappear, 45 have emergency 
locators, and only 3 are not discovered. The "conditional" probability 
that a lost plane with an emrgency locator will not be discovered is 
3/45.

>b. If it does not have an emergency locator, what is the probability 
    that it will be discovered?

55 planes that get lost don't have emergency locators. 28 are 
discovered. 

You can take it from there.

-Doctor Gary,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/