Permutations
A permutation refers to
one possible ordering of a set (or subset) of objects.
For example, if you
have a red ball, a green ball, and a blue ball, one possible way of
ordering these three is: red, green, blue. This is one permutation.
Another permutation would be blue, red, green.
We can choose the
first ball in one of three ways; once we’ve made this choice
then that ball can no longer be chosen. Now there are only two ways to
choose the second ball. Finally we’re left with one way to choose
the third ball.
3 * 2 * 1 =
6 permutations.
This multiplication
that involves all the numbers between 1 and 3 is called a factorial
and is written 3!
The following
permutation tree is helpful to visualize these choices. The node
labeled 1 is the root of the tree. At that node we haven't made a
choice yet, however we have 3 to choose from (3 is in the top node to
signify this.) Once we made a choice, we are at the second level.
Notice the colors of the interconnecting lines. These colors signify
the ball we chose. Once you make a choice you have committed
yourself to that branch of the tree. Here you see there are 3 main
branches, one for each color. The next node contains the number 2
which means we have 2 possible choices to make. The first choice
cannot be chosen again. If we had multiple
balls of the same color then they would act as the other balls as
far as choices are concerned. Finally we have only one choice to make,
signified by 1 inside the node. The nodes containing letters are the
terminating nodes, sometimes called leaf nodes, and represent each
permutation possible.
Following the nodes from the root to the leaves we have the following
6 permutations:
a: red blue green
b: red green blue
c: blue red green
d: blue green red
e: green red blue
f: green blue red
Probability of Independent
Events
Independent events are
events having no effect on each other. We say that there is
no history (or memory) from event to event. If I roll a die two times
then the outcomes of these events are completely unrelated. When we talk about the
probability of independent events occurring in a particular order (Or sequence,) we
calculate each event probability, and then multiply all probabilities to get the
probability of that sequence.
Example:
Toss a coin.
What is the probability it will be heads
The event space (or
sample space) is 2, 1 head, and 1 tail. There is only one way to
land heads. So the number of permutations to get heads is 1. To get the
probability we divide this number of permutations by the number of
choices in the sample space, 2, and get the probability
of the coin being heads: ½.
Example:
Toss a coin twice in a
row. What is the probability it will be heads both times?
Well, the probability
of the coin being heads is ½. This is the same for the
second toss. These are two unrelated events but are in sequence
so we multiply to get ½ · ½ = ¼ = 0.25
= 25%.
Example:
A coin is tossed 6
times and comes up heads each time. What is the probability the next
toss will be heads?
These are independent
events which have no history, so the previous 6 tosses are irrelevant
(we were not asked for a given sequence, i.e, 7 heads in a row), The
answer is therefore 50%.
Example:
A spinner on a
particular board game is on the center of a circle divided into 10
equal sectors. The spinner is spun twice. What is the probability
that on the first spin it stops on a 6 or a 3, then on the second
spin it stops on the 10?
Well on the first spin
we have 2 desired outcomes out of 10, 2/10 = 0.20 and the second
spin we have 1 desired outcome out of 10, so we have 1/10 = 0.1.
Since this is a sequence
of independent events we multiply and the probability that
this sequence will occur is (0.2)(0.1) = 0.02 = 2%.
Side note: many casinos
maintain and show on a monitor the outcomes of ongoing rolls of a roulette
table. The chances of the roulette ball landing on any given slot is the same as
for any other slot, and each roll is independent of the previous roll. Many are
falsely led to believe that this "history" will give them a better chance
knowing which color and/or number will occur next and therefore be the winning bet.
Now, you know better; there is no history for these outcomes.
This is another FREE ALGEBRA PRINTABLE presented to you from the
Algebra section of
K12math.com