Coin Flipping

Date: 01/26/98 at 05:19:38
From: Lee, Choong Loon
Subject: Probability

I wonder how I can figure out the chances of the following case:

Flipping a coin five times with the result a combination of T,T,T,H,H.
(T=Tail, H=Head) 

To flip the same coin five times, what will be the chances of getting 
the same combination (exact sequence) "right away"? I know that
1/5X1/5X1/5X1/5X1/5 is the formula to get 3T and 2H right away. 
Somebody told me there is some method call "condition" special for 
this kind of problem. It is like 1/2 X1/3 X1/4 X1/5 X1/6 but I am not 
sure. 

Please help me solve this case and if possible please explain in 
detail. Thank you very much.

Lee Choong Loon

Date: 02/01/98 at 22:10:37
From: Doctor Wolf
Subject: Re: Probability

Let's begin by reviewing The Fundamental Principle of Counting, a 
mainstay of probability theory:

If experiment A can result in N distinct outcomes, and experiment B 
can result in M distinct outcomes, then experiment A followed by 
experiment B can result in exactly N*M different outcomes. And of 
course this can be generalized to more than two experiments.

Example: You roll a single die, then flip a coin. How many different
outcomes are possible? Let's see... 6 outcomes for the die, 2 for the
coin, so 6*2 or 12 different outcomes are possible. They may be
thought of as (3,T), (1,H), (6,H), etc.  

Also, since the outcome of the die in no way affects the tossing of 
the coin, each of the 12 possible outcomes will have probability 
(1/6)*(1/2) or 1/12. This is called an "equiprobability space," and is 
quite common.

Now, back to your problem. You will be performing the same experiment
5 times in succession; that is, flipping a coin. Each flip of the 
coin can result in 2 distinct and equally likely outcomes, H or T. 
Moreover, the result of any coin flip is not influenced by or 
dependent upon any previous coin flip. That last statement regarding 
independence of the coin flips is very important; it tells us that all 
possible outcomes after 5 coin flips are equally likely, or have the 
same probability.

By the Principle of Counting, there are 2*2*2*2*2, or 32 possible
outcomes to your problem. The one you are interested in is (TTTHH).
The chance of getting T(first flip), then T(second flip), then T(third 
flip), H on the fourth flip and H on the fifth flip is:

             (1/2)*(1/2)*(1/2)*(1/2)*(1/2) = 1/32.

In fact, every one of the 32 possible outcomes from flipping a coin 5 
times has the same probability .... 1/32.

Hope this helped .... drop in again soon!

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