Conditional Probability

Date: 07/18/98 at 22:44:54
From: Carole Black
Subject: Conditional Probability

I will be a beginning math teacher in the fall and will be teaching 
Statistics. I am "boning up" on conditional probabilities and I have 
a question about an example in the Basic Probability information from 
the Ask Dr. Math faq. The example is discussing the independent 
events of drawing red or blue marbles. There are 6 blue marbles and 4 
red marbles. The discussion goes on to talk about two events, the 
second outcome dependent upon the first. The actual example is: But 
suppose we want to know the probability of your drawing a blue marble 
and my drawing a red one? 

Here are the possibilities that make up the sample space:

   a. (you draw a blue marble and then I draw a blue marble) 
   b. (you draw a blue marble and then I draw a red marble) 
   c. (you draw a red marble and then I draw a blue marble) 
   d. (you draw a red marble and then I draw a red marble) 

The calculation for b is given as:

   your probability of drawing a blue marble (3/5) multiplied 
   by my probability of drawing a red marble  (4/9):
   
   3/5 x 4/9 = 12/45 or, reduced, 4/15. 

My question is: is this the same thing as P(Red|Blue)? I believe 
these are two different things, but I am confused as to how to explain 
the difference. For P(Red | Blue) I calculate this probability as:

   (4/15)/(6/10) = 4/9.  

Can you help clear up my confusion so I can explain this clearly to my 
students in the fall?

Thank you,
Carole Black

Date: 07/19/98 at 08:07:58
From: Doctor Anthony
Subject: Re: Conditional Probability

Your second answer P(Red|Blue) = 4/9 is correct

This means that the probability of drawing a Red given that the first 
draw was a blue is 4/9. Note the word 'given'. We know before making 
the second draw that the first draw was a blue. This must be contrasted 
with the probability of red-blue before we start making any draw. The 
probability of red-blue is 6/10 x 4/9 = 4/15 and this probability is 
calculated before the result of the first draw is known.

The word 'conditional' alerts us to the fact that we are calculating 
probabilities 'conditional' on knowing further information partway 
through the experiment. These probabilities are also referred to as 
'Bayesian' probability, named after the probability theorist Thomas 
Bayes (1702-61) who gave this theorem:

             P(A and E)
   P(E|A) =  ----------
                P(A) 

In other words, if we know that A has occurred, then the sample space 
is reduced to the probability of event A, and the denominator for 
P(A and E) is not 1 but P(A).

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

Date: 07/19/98 at 18:50:40
From: Carole Black
Subject: Conditional Probability

Dr. Anthony, thank you for your very quick and wonderfully clear 
explanation to my question.   

Carole Black