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Unit 4 – Randomness and Probability
(Chapter 15 –Probability Rules)
Key Terms:
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Sample space –
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Disjoint events –
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Addition rule –
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The most common kind of picture to make is called a __________________.
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When two events A and B are disjoint, we can use the addition rule for disjoint
events from Chapter 14:
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However, when our events are not __________________, this earlier addition
rule will double count the probability of both A and B occurring. Thus, we need
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General Addition Rule –
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Conditional Probability –
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Independence (casual) –
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Multiplication rule –
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General multiplication rule –
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Independence (formal) –
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Tree diagram
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For any random __________________, each trial generates an outcome.
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An event is any set or collection of __________________.
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The collection of all possible outcomes is called the __________________
the __________________.
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General Addition Rule:
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denoted S.
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When outcomes are __________________, probabilities for events are easy to
find just by counting.
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For any two events A and B,
The following Venn diagram shows a situation in which we would use the
general addition rule:
A survey of high school students revealed that 54% are enrolled in a Calculus class,
32% are enrolled in a Statistics class, and 22% are enrolled in both classes.
a.
What is the probability a student is in Calculus or Statistics?
b.
What is the probability that a student was enrolled in only a
When the k possible outcomes are equally likely, each has a probability of
1/k.
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For any event A that is made up of equally likely outcomes,
Calculus class?
c.
What is the probability that a student selected at random was
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not enrolled in either a Calculus or a Statistics class?
d.
When two events A and B are independent, we can use the multiplication rule
for independent events from Chapter 14:
Are being in a Calculus class and being in a Statistics class disjoint
events? Explain.
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However, when our events are not independent, this earlier multiplication rule
does not work. Thus, we need the ____________________________________.
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Back in Chapter 3, we looked at contingency tables and talked about
conditional distributions.
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We encountered the general multiplication rule in the form of
__________________ probability.
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Rearranging the equation in the definition for conditional probability, we get
When we want the probability of an event from a conditional distribution, we
the General Multiplication Rule:
write P(B|A) and pronounce it “____________________________________.”
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For any two events A and B
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Or
A probability that takes into account a given condition is called a conditional
probability.
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To find the probability of the event B given the event A, we restrict our
attention to the outcomes in A. We then find the fraction of those
outcomes B that also occurred.
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Independence of two events means that the outcome of one event does not
influence the probability of the other.
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Note: P(A) cannot equal 0, since we know that A has occurred.
With our new notation for conditional probabilities, we can now
formalize this definition:
• Events A and B are independent whenever P(B|A) = P(B).
(Equivalently, events A and B are independent whenever P(A|B)
= P(A).)
Use the chart to calculate the following conditional probabilities:
1.
P(married | age 18 to 24)
2.
P(widowed | at least 65)
Jeans
Other
Total
Male
12
5
17
Female
8
11
19
Total
20
16
36
1.
What is the probability a male wears jeans?
2.
What is the probability that someone wearing jeans is a male?
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We often sample without replacement, which doesn’t matter too
much when we are dealing with a large population.
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However, when drawing from a small population, we need to take
note and adjust probabilities accordingly.
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Drawing without replacement is just another instance of working with
conditional probabilities.
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A _______________________________- helps us think through conditional
probabilities by showing sequences of events as paths that look like branches
of a tree.
3.
Are gender and attire independent?
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Making a tree diagram for situations with conditional probabilities is
consistent with our “make a picture” mantra.
EXAMPLE:
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Disjoint events cannot be independent! Well, why not?
1.
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
In April 2003, Science Magazine reported on a new computer-based test for ovarian
cancer that examines a blood sample for the presence of certain patterns of
proteins. Ovarian cancer, though dangerous, is very rare, afflicting only 1 of every
5000 women. The test is highly sensitive, able to correctly detect the presence of
2.
_____________________________________________________________
ovarian cancer in 99.97% of women who have the disease. However, it is unlikely
_____________________________________________________________
to be used as a screening test in the general population because the test gave false
_____________________________________________________________
positives 5% of the time. Why are false positives such a big problem? Draw a tree
diagram and determine the probability that a woman who tests positive using this
3.
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
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A common error is to treat disjoint events as if they were independent, and
apply the Multiplication Rule for independent events—don’t make that
mistake.
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Sampling __________________________________________ means that once
one individual is drawn it doesn’t go back into the pool.
method actually has ovarian cancer.