Probability by 4VtD7tW



 Parts of life are uncertain. Using
notions of probability provide a way
    to deal with the uncertainty.

Probability is a numerical measure of the likelihood that an
‘event’ will occur.
Probability values range from 0 to 1. 0 means the event will not
occur. 1 means the event will surely occur. .5 means that an
event is as likely to occur as not occur.
The closer the probability is to 1, the more likely the event is to
Note in the definition of probability the word event is used. We
will come back to the word event, but now we turn to the idea of
an experiment.
An experiment is a process that generates well defined
outcomes, or what we call sample points.

Some examples of experiments are:
- toss a coin, with sample points being heads, tails.
- conduct a sales call, with sample points being sale, no sale.
The sample space for an experiment is the set of all
experimental outcomes.
In order to assign probabilities one has to be able to count all
the experimental outcomes. Next let’s study some rules to
help us think about how to count all the experimental

Multiple-Step Experiments
If an experiment can be described as a sequence of k steps with
n1 possible outcomes on the first step, n2 possible outcomes on
the second step, and so on until we get to nk, then the total
number of experimental outcomes is the product (n1)(n2)...(nk).
Say a construction project has two stages - design and
construction. If the design could be completed in 2, 3, or 4
months and the construction could be done in 6, 7, or 8 months,
then there are (3)(3) = 9 different experimental outcomes. I will
list the outcomes as ordered pairs of numbers, with the first
number the time to complete the design and the second the time
to complete the construction: (2, 6), (2, 7), (2, 8), (3, 6), (3, 7),
(3, 8), (4, 6), (4, 7), and (4, 8).

Another type of experiment consists of taking n objects from a
larger set of N objects. An example of this might be a case
where there are 5 job applicants, but only 2 jobs. How many
different experimental outcomes are there? (by the way some
outcomes may have a greater chance of occurring because some
of the people may have more favorable characteristics - but here
we just want to list possible outcomes.)
We will use factorial notation here - this is a math thing. For
example, 5! = (5)(4)(3)(2)(1) =120
The formula for the number of combinations when taking n from
N is N!/[n!(N-n)!]. For example from above
  5!         = (5)(4)(3)(2)(1) =      (5)(4)     = 10
2!(5 - 2)!     2!(3)(2)(1)              2                         5
Remember having a lock at school? The dial on the lock might
have had 40 numbers. To open the lock you spun the dial to the
right several times and settled on the first number in the
combination, then you went around to the left once past that
number and then settled on the second number, then you went
right to the third number.
Say you had the combination 7 - 16 - 32. This is 3 numbers
from 40. Is the combination 32 - 7 - 16 the same as 7 - 16 -32?
The answer is no. Parker, what is the point? The idea of a
combination on the previous screen dealt with combinations
where order did not matter. 7 - 16 - 32 and 32 - 7 - 16 would be
the same and counted once. But in a permutation they are
different. Perhaps a better name for our locks would be       6
permutation locks – order matters.
The formula for the number of permutations when taking n from
N is N!/(N-n)!. For example 2 from 5 is
  5!        = (5)(4)(3)(2)(1) =     (5)(4)     = 20
(5 - 2)!           (3)(2)(1)
Let’s do another example. Say we have the letters A, B, and C.
Say we want to choose 2 of these. We have
3!/(3-2)! = 3(2)/1 = 6.
The permutations would be
AB, BA, AC, CA, BC, CB, but combinations are
So, there are less combinations than permutations.
Assigning Probabilities
Each experimental outcome must be assigned a probability of 0,
or 1, or somewhere between 0 and 1.
The other feature of assigning probabilities is that the sum of the
probabilities of all the experimental outcomes must be 1.
There are three ways of assigning probabilities: the classical
method, the relative frequency method and the subjective method.
The classical method is used when each of the experimental
outcomes is equally likely to occur. If there are t experimental
outcomes, then the probability of any one experimental outcome
is 1/t. An example is a roll of a die. There are 6 possible
outcomes and each one has probability 1/6.

The relative frequency method is used when data is available
about the past history of the experiment. The probability of an
outcome is the relative frequency of the outcome. You can
form this probability by taking the ratio of the number of times
the outcome came up over the total number of times of the
experiment. As an example, if out of 100 sales calls, you had
37 sales, the probability of a sale would be 37/100 = .37.
The subjective method is used when the other two can not be
used. You and a friend may consider the experiment that
Nebraska will win the national championship in football this
year. The outcomes are either they will win or they will not
win. You may say P(win) = .4 (meaning the probability of a
win is .) Thus you also say P(not win) = .6. Your friend might
say the probabilities should be .7, .3, respectively. Your
differences represent your subjectivity.
Let’s do a problem. Say we have a population of 50 bank
accounts and we want to take a random sample of four accounts in
order to learn about the population. How many different random
samples of four accounts are possible?
Since we do not have a situation like a ‘permutation’ lock (once
account is chosen, its in), we use the combination formula:
 50!          =     (50)(49)...(2)(1)   = (50)(49)(48)(47)
4!(46!)           4!(46)(45)...(2)(1)         (4)(3)(2)(1)

= 230300.
Let’s do another example on the next slide.

Say you have an ordinary deck of playing cards - you know, ace
through king in spades, hearts, diamonds, and clubs. So there are
52 cards in the deck. In many games of poker you get 5 cards.
How many different combinations of 5 cards are there?
52!       = (52)(51)(50)(49)(48) = 2598960
5!(47!)        (5)(4)(3)(2)(1)
This means there are 2 million, 598 thousand, 960 different
combinations of hands you could be dealt. Most hands you get
are not memorable - you know, you get a 7 of hearts, queen of
spades, 3 of clubs, 9 of clubs and a 4 of spades. But a royal flush
hearts - 10, jack, queen, king, and ace all hearts - is memorable.
Each hand mentioned has a 1 divided by 2598960 chance of
happening. But the royal flush is a winner!


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