Transcript Procedure Example of event

Chapter 4
Probability
4-1 Overview
4-2 Fundamentals
4-3 Addition Rule
4-4 Multiplication Rule: Basics
4-5 Multiplication Rule: Complements and
Conditional Probability
4-6 Probabilities Through Simulations
4-7 Counting
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Section 4-1
Overview
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Overview
Rare Event Rule for Inferential Statistics:
If, under a given assumption, the probability of a
particular observed event is extremely small, we
conclude that the assumption is probably not correct.
Statisticians use the rare event rule for inferential
statistics.
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Example
• We have a gender selection procedure that
claims to increase the birth of baby girls.
• Actual results out of 100 babies born: 98 girls,
2 boys.
• Is there a chance of getting boys?
• What’s the probability?
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Section 4-2
Fundamentals
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Key Concept
This section introduces the basic
concept of the probability of an event.
Three different methods for finding
probability values will be presented.
The most important objective of this
section is to learn how to interpret
probability values.
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Example
• We have a quality control test that shows that
there are 5 defective printers and 15 that are
“good.”
• What is the probability of choosing one that is
not defective?
• What is the probability of choosing one that is
defective?
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Definitions
 Event
any collection of results or outcomes of a
procedure
 Simple Event
an outcome or an event that cannot be further
broken down into simpler components
 Sample Space
for a procedure consists of all possible simple
events; that is, the sample space consists of all
outcomes that cannot be broken down any
further
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Example
• In the following display, we use f to denote a
female baby and m to denote a male baby.
Procedure
Example of event
Single birth
female
(simple event)
3 births
2 females and 1 male
(not a simple event)
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Example
• In the following display, we use f to denote a
female baby and m to denote a male baby.
Procedure
Example of event
Sample Space
Single birth
female
{f, m}
(simple event)
3 births
2 females and 1 male {fff, ffm, fmf,
(not a simple event) fmm, mff,
mfm, mmf,
mmm}
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Notation for
Probabilities
P - denotes a probability.
A, B, and C - denote specific events.
P (A) -
denotes the probability of
event A occurring.
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Basic Rules for
Computing Probability
Rule 1: Relative Frequency Approximation
of Probability
Conduct (or observe) a procedure, and count
the number of times event A actually occurs.
Based on these actual results, P(A) is
estimated as follows:
P(A) =
number of times A occurred
number of times trial was repeated
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Example
Find the probability that NBA basketball player
Reggie Miller makes a free throw after being
fouled. At one point in his career, he made
5915 free throws in 6679 attempts (NBA
data).
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Activity
• Flip a penny 30 times. Record your results on
the board for # of heads flipped. Let A = # of
heads flipped.
Trial #
Heads
Tails
P(A)
• Plot your data (x-axis: trial number, y-axis:
proportion)
What do you notice about the graph?
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Law of
Large Numbers
As a procedure is repeated again and
again, the relative frequency probability
(from Rule 1) of an event tends to
approach the actual probability.
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Basic Rules for
Computing Probability - cont
Rule 2: Classical Approach to Probability
(Requires Equally Likely Outcomes)
Assume that a given procedure has n different
simple events and that each of those simple
events has an equal chance of occurring. If
event A can occur in s of these n ways, then
s
P(A) = n =
number of ways A can occur
number of different
simple events
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Example
1) If you have four kings from a deck, what’s the
probability you’ll get a king of diamonds?
2) If trying to determine P(2) with a fair and balanced
die, each of the six faces has an equal chance of
occurring. Find P(2).
3) Find the probability that when a couple has 3
children, they will have exactly 2 boys. Assume that
boys and girls are equally likely and that the gender
of any child is not influenced by the gender of any
other child.
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Basic Rules for
Computing Probability - cont
Rule 3: Subjective Probabilities
P(A), the probability of event A, is estimated
by using knowledge of the relevant
circumstances.
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Examples
1) When trying to estimate the probability of
rain tomorrow, meteorologists use their
expert knowledge of weather conditions to
develop an estimate of the probability.
2) What is the probability that your car will
be hit by a meteorite this year?
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Probabilities/Outcomes that aren’t
equally likely
• Is the probability that a Republican will win
the next presidential election 1/2 ? In other
words, is the probability that a Republican will
win equal to the probability that a Democrat
will win?
• Is there a ½ probability that you will pass your
next statistics test?
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Warning!
When you know nothing about the likelihood
of different possible outcomes, don’t assume
they are equally likely!
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Probability Limits
 The probability of an impossible event is 0.
 The probability of an event that is certain to
occur is 1.
 For any event A, the probability of A is
between 0 and 1 inclusive.
That is, 0  P(A)  1.
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Possible Values
for Probabilities
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Example
•
If a year is selected at random, find the
probability that Thanksgiving Day will be on
a:
a) Wednesday
b) Thursday
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Definition
The complement of event A, denoted by
A, consists of all outcomes in which the
event A does not occur.
Example: In reality, more boys are born
than girls. In one typical group, there are
205 newborn babies, 105 of whom are
boys. If one baby is randomly selected
from the group, what is the probability
that the baby is not a boy?
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Rounding Off
Probabilities
When expressing the value of a probability,
either give the exact fraction or decimal or
round off final decimal results to three
significant digits. (Suggestion: When the
probability is not a simple fraction such as 2/3
or 5/9, express it as a decimal so that the
number can be better understood.)
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Definitions
The actual odds against event A occurring are the ratio
P(A)/P(A), usually expressed in the form of a:b (or “a to
b”), where a and b are integers having no common
factors.
The actual odds in favor of event A occurring are the
reciprocal of the actual odds against the event. If the
odds against A are a:b, then the odds in favor of A are
b:a.
The payoff odds against event A represent the ratio
of the net profit (if you win) to the amount bet.
payoff odds against event
A = (net profit) : (amount bet)
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Odds! (Vegas, baby!)
If you bet $5 on the number 13 in roulette, your
probability of winning is 1/38 and the payoff
odds are given by the casino as 35:1.
a) Find the actual odds against the outcome of
13.
b) How much net profit would you make if you
win by betting on 13?
c) If the casino were operating “just for the fun
of it” (yeah, right), and the payoff odds were
changed to match the actual odds against 13,
how much would you win if the outcome
were 13?
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Recap
In this section we have discussed:
 Rare event rule for inferential statistics.
 Probability rules.
 Law of large numbers.
 Complementary events.
 Rounding off probabilities.
 Odds.
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