Probability and Pascal's Triangle

Date: 06/04/98 at 22:27:07
From: Teresa Ramirez
Subject: Probability (Pascal's Triangle)

How do I find the probability of getting 3 heads and 2 tails using 
Pascal's Triangle?

Date: 06/05/98 at 00:03:14
From: Doctor Pat
Subject: Re: Probability (Pascal's Triangle)

Teresa,

Pascal's triangle is a great tool for this kind of problem, as long as 
you remember that it only works if the probability of both things is 
the same on each try. If you had a bent coin so that heads came up 
with a probability of 1/4 or something, it would not work.  

With that out of the way, let's talk about what can happen when we 
flip coins, and what Pascal's triangle can offer in help.  

If I flip one coin, it has two possible outcomes, 0 heads or 1 head. 
If we look at the first row down from the point on Pascal's triangle 
we notice that 1  1 show up and that they add up to two, giving 1/2 
for the probability of no heads and 1/2 for the probability of one 
head.

For the next row we see that the numbers are:

   1      2      1    

and these three numbers add up to four. If we think about flipping a 
coin we can get no heads, we can get one head in two different ways 
(h,t or t,h) and we can get two heads in only one way. Again we notice 
that the probability of getting any number of heads is the same as a 
number in this row over the total of the row.  
    
What about for three flips? Pascal's third row gives:

   1      3      3      1   

for a total of 8 outcomes. Flipping a coin gives:
  
   ttt             >>>>>>>>>>>>      1       way to get 0 heads 
   htt, tht, tth   >>>>>>>>>>>>      3       ways to get 1 head
   hht, hth, thh   >>>>>>>>>>>>      3       ways to get 2 heads
   hhh             >>>>>>>>>>>>      1       way to get 3 heads 

Thus, the probability of getting each is:

   P(0 heads) = 1/8   
   P(1 head)  = 3/8
   P(2 heads) = 3/8
   P(3 heads) = 1/8

Now you wanted to know about flipping five coins. The fifth row of 
Pascal's triangle looks like:

   1      5      10      10      5      1   

for a total of 32 possible outcomes. Can you find the probability of 3 
heads (and two tails)? Can you write down all the orders in which they 
occur?  Good luck, I hope I've been able to help you see how Pascal's 
triangle is a handy tool for some types of probability problems.

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