Transcript Elementary Probability Theory

4.3 Trees and Counting Techniques
This rule extends to outcomes involving three, four, or more series of events.
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Example
• A student needs to take Math 2260 and English
2201 in the summer of 2014. If there are 4
sections of Math 226 and 3 sections of English
2201 to choose, how many different ways are
there to select two courses?
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Multiplication Rule for Counting
• Consider the series of event E1 through Em,
where n1 is the number of possible outcomes for
event E1, n2 is the number of possible outcomes
for event E2, and nm is the number of possible
outcomes for event Em.Then the product
n1xn2x…..xnm
gives the total number of possible outcomes for
the series of events E1, followed by E2, up through
Em
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Example
• A student needs to take Math 2260, English
2201, and CSCI 1301 in the summer of 2014. If
there are 4 sections of Math 2260, 3 sections of
English 2201, and 2 sections of CSCI 1301 to
choose, how many different ways are there to
select two courses?
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Tree Diagrams
• Displays the outcomes of an experiment
consisting of a sequence of activities.
– The total number of branches equals the
total number of outcomes.
– Each unique outcome is represented by
following a branch from start to finish.
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Tree Diagrams with Probability
• We can also label each branch of the tree with
its respective probability.
• To obtain the probability of the events, we can
multiply the probabilities as we work down a
particular branch.
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Urn Example
•
Suppose there are five balls in an urn. Three are red and two are
blue. We will select a ball, note the color, and, without replacing the
first ball, select a second ball.
There are four
possible outcomes:
Red, Red
Red, Blue
Blue, Red
Blue, Blue
We can find the
probabilities of the
outcomes by using
the multiplication rule
for dependent events.
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More Examples
• How many different ways are there to arrange 3
people in a line?
• How many different ways are there to arrange 4
people in a line?
• If a chair and a vice chair are selected from a
committee of 6, how many different ways are
there to make the selection?
• If a chair, a vice chair, and a sectarary are
selected from a committee of 6, how many
different ways are there to make the selection?
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Factorials
• For counting numbers 1, 2, 3, …
• ! is read “factorial”
– So for example, 5! is read “five factorial”
• n! = n * (n-1) * (n-2) * … * 3 * 2 * 1
– So for example, 5! = 5 * 4 * 3 * 2 * 1 = 120
• 1! = 1
• 0! = 1
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Permutations
• Permutation: ordered grouping of objects.
• Counting Rule for Permutations
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Examples
• If two representatives are selected from a
committee of 6, how many different ways are
there to make the selection?
• If three representatives are selected from a
committee of 6, how many different ways are
there to make the selection?
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Combinations
• A combination is a grouping that pays no
attention to order.
• Counting Rule for Combinations
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More Example
• What is the probability to win the jackpot of
Mega million lottery? (5 out of 75 and 1 out of
15)
• What is the probability to win the jackpot of
power ball lottery? (5 out of 59 and 1 out of 35)
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Practice
a) Make a tree diagram to show all the possible
sequences of answers for three
multiple-choice questions, each with four possible
responses.
(b) Assuming that you are guessing the answers
so that all outcomes listed in the tree are equally
likely, what is the probability that you will guess the
one sequence that contains all three correct
answers?
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Practice
In the Cash Now lottery game there are 10 finalists
who submitted entry tickets on time. From these
10 tickets, three grand prize winners will be drawn.
The first prize is one million dollars, the second
prize is one hundred thousand dollars, and the
third prize is ten thousand dollars. Determine the
total number of different ways in which the winners
can be drawn. (Assume that the tickets are not
replaced after they are drawn.)
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