Understanding Probability

Date: 05/13/2002 at 03:27:22
From: LuckyStar
Subject: Mathematics - Probabilities

I am doing a science project.  If I were looking for a percentage 
or probability (e.g. for colored marbles) how would I go about it?

To say it in a different way, say I were looking for a 
probability of colored marbles in three packages.  How would I do 
that?

Date: 05/13/2002 at 10:10:49
From: Doctor Ian
Subject: Re: Mathematics - Probabilities

Hi,

The general idea is this.  You start by defining a set of things 
that can happen, and identifying a subset of those things that 
are of interest to you. 

For example, suppose I flip a coin three times.  If I use H to 
represent heads and T to represent tails, here are all the things 
that might happen:

  1st      2nd       3rd 
  toss     toss      toss

  H     -> HH     -> HHH
                     HHT
       
           HT     -> HTH
                     HTT

  T     -> TH     -> THH
                     THT

           TT     -> TTH
                     TTT

In other words, there are 8 possible outcomes:

  HHH, HHT, HTH, HTT, THH, THT, TTH, TTT

Now, suppose I decide that I'm interested in tosses that result 
in exactly two heads.  How many of those are there? 

  HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
       ---  ---       ---     

There are three.  So the probability of getting exactly two heads 
is

      number of ways to get two heads
  p = -------------------------------------
      number of ways to get anything at all

       3
    = ---
       8

Now, what if I decide that I'm interested in exactly two of 
_either_ heads or tails.  How many ways can that happen? 

  HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
       ---  ---  ---  ---  ---  ---   

There are 6.  So the probability is

      number of ways to get two heads or two tails
  p = --------------------------------------------
      number of ways to get anything at all

       6
    = ---
       8

So this is the basic idea behind probability.  Where it gets 
tricky is this:  As you start to involve more objects and more 
events, the numbers grow very quickly.  For example, the number 
of things that can happen when you flip a coin N times is

  N    Number of possibilities
  --   -----------------------
   1   2^1  =             2
   2   2^2  =             4
   3   2^3  =             8
   4   2^4  =            16
   5   2^5  =            32
   6   2^6  =            64
  10   2^10 =         1,024
  20   2^20 =     1,048,576
  30   2^30 = 1,073,741,824

And these are relatively small numbers, because a coin has only 
two sides it can land on.  Suppose we roll a die N times.  How 
many sequences can we get? 

  N    Number of possibilities
  --   -------------------------------------
   1   6^1 =                               6
   2   6^2 =                              36
   3   6^3 =                             216
   4   6^4 =                           1,296
   5   6^5 =                           7,776
   6   6^6 =                          46,656
  10   6^10 =                     60,466,176
  20   6^20 =          3,600,000,000,000,000  (approximately)
  30   6^30 = 22,000,000,000,000,000,000,000  (approximately)

Because the numbers grow so quickly, it becomes impossible to 
actually list all the possible events for all but the most 
trivial situations.  Hence all the interest in coming up with 
formulas (like the ones you can find in our FAQ on "Permuations 
and Combinations") that can be used to compute probabilities 
directly!

Also, probability often involves the use of tricks.  One common 
trick is this:  Sometimes it's easier to figure out how many ways 
something _can't_ happen than to figure out how many ways it 
_can_ happen. You can find an example of that here:

  http://mathforum.org/dr.math/problems/murad.04.03.02.html 

Because the numbers in probability grow so quickly, when you 
can't find a formula that matches your particular problem, often 
the best thing to do is play around with smaller versions of your 
problem (e.g., instead of using 20 marbles, use 2 or 3 or 4), 
looking for some kind of pattern that you can use to create a 
formula that you can use for the larger case. 

I hope this helps.  Write back if you'd like to talk more
about this, or anything else.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/