MBAD 51415142

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MBAD 5141/5142

• Probability Part II

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Outline for todays class

• Test Analysis
• Homework Help
• Combinations/Permutations
• Random Variables
• Probability Distributions

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Combinations and Permutations
Since probability is contingent on the ratio of
successful events to sample space it can be seen
that the ability to count the different
occurrences becomes important. It is often the
case that both the sample space and number of
successes is quite large. For example, if you
were interested in the event rolling at least
two sixes in six rolls of a die then one part of
the solution would be to determine the sample
space. Since there are six outcomes in one roll
of a die and six rolls there are 6x6x6x6x6x6 or
46656 possible outcomes. These would be difficult
to list.
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Combinations and Permutations (continued)
In mathematics there are many ways to count large
numbers. One way is to count the combinations.
For example, a company wants to send sales
representatives to three locations. The company
has ten representatives that can be sent. In how
many ways can three be chosen? This is an example
of a combination and can be written symbolically
as
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Combinations and Permutations (continued)
So, if we continue to use the example then we
want to know how many combinations of three can
we make out of ten.
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Combinations and Permutations (continued)
In the previous example, the fact that we
repeated numbers did not matter. If we were to
list out the possible combinations we could have
listed 1,2,3 or 2,3,1. In both cases the numbers
1, 2, and 3 are used but they are unique
combinations. What if repeats were not
acceptable? For example, In a group of four
people we are required to pick persons to
participate in three different tests. In how many
ways can the selection be made? Assume that a
person can only take one test. What could be some
possible ways to pick?
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Combinations and Permutations (continued)
If we pick person 1 to take test A, then he
cannot be picked to take test B. Therefore, only
persons 2,3 and 4 can be selected for test B. If
person 2 is selected for test B, then she cannot
take test C. therefore only person 3 and 4 can be
selected for test C. Can you tell by now that the
number of ways to pick the people can become
complicated. When no repeats are allowed in the
picking, then we have a way to select through
Permutation.
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Combinations and Permutations (continued)
PERMUTATIONS are symbolized as follows
So, if we picked three out of four to take the
test and repeats are not allowed,
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Combinations and Permutations (continued)
It is interesting to note here that both 1! and
0! are both 1.
For homework tonight in this topic do 1,7, and 9
on page 377. Do not do the problems listed in the
syllabus from this section.
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Random Variables
Consider, again, the experiment of tossing two
coins. We already know that the possible outcomes
are HH, HT, TH, and TT. What if we are interested
in the number of heads in each outcome. Then the
outcome HH would be 2, the outcome HT would be 1,
the outcome TH would be 1, and the outcome TT
would be 0. This is an example of a random
variable. In other words, the number of heads in
each toss of two coins varies from 0 to 2. This
idea, then can be formalized into a definition
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Random Variables (continued)
A RANDOM VARIABLE is a variable that assumes
numerical values associated with the random
outcomes of an experiment, where one (and only
one) numerical value is assigned to each sample
point. These variables are normally denoted with
capital letters (i.e., X or Y) It is important
to note here that the variable is random and can
take on assigned values at random.
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Random Variables (continued)
Consider the following examples 1)A couple has
five children and you are interested in the
number of boys. Define the random variable. 2)
Upon receiving an insurance premium a company has
four months to pay. You are interested in the
number of weeks before the premium is paid.
Define the random variable. (assume four weeks in
a month) 3) Six athletes are competing a a 100
meter dash. You are interested in the time it
takes all to finish. Define the random variable.
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Random Variables (continued)
Now you can see that the number of ways the
variable can occur depends on the situation.
Since this is so we divide random variables into
two categories DISCRETE and CONTINUOUS. DISCRETE
random variables are countable. An example of
this is 1 and 2 in the previous slide. A
CONTINUOUS random variable take place over
intervals and are not countable. An example of
this is 3 in the previous slide. Look at 4.3,
page 565 in your text.
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Probability Distributions
Random variables and their associated
probabilities can be put in tabular for. These
are know and PROBABILITY DISTRIBUTIONS. Take our
two coin toss experiment
It is always assumed that each individual
probability p(x) is never 0 and the sum of all
probabilities in the distribution is 1. These
distributions can also be illustrated in
Histograms of Stemplots.
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Probability Distributions (Histogram)
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Probability Distributions (means)
Since we can now make a distribution out of a
random variable we can begin to formulate some
basic analyses of these distributions. In
statistics, it is always good to know here the
data cluster or center. We can find where the
random variable distribution centers by finding
its mean. The formula is
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Probability Distributions (means)
So, the mean of our coin toss distribution is
The mean (or expected) value of our distribution
is 1. Put in context, if we were to toss two
coins many times over and over we would expect
the number of heads that appear in each toss to
average 1.
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Probability Distributions (means)
If it costs 1 to play the PICK THREE lottery in
NC what would be your expected payout? If you are
not familiar with this game, it is played by
picking any three digit number and if your digits
match the winning ones in any order you win 100.
So, if the lottery picks the number 437 and you
have picked a 374, then you win. Each digit can
be picked from the numbers 0 through 9.
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Probability Distributions (variance)
It is also helpful to determine how far away from
the center certain variables or probabilities
lie. To find this we must find the variance
The STANDARD DEVIATION is the square root of the
variance
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Variance
Find the standard deviation of the above
distribution.
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Probability Determination
Find P(X3) Find P(Xlt2) Find P(X4)