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### 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:

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

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations
High School Probability
Middle School Probability

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