Transcript AP Stat 5.2 PP

CHAPTER 5
Probability: What Are
the Chances?
5.2
Probability Rules
The Practice of Statistics, 5th Edition
Starnes, Tabor, Yates, Moore
Bedford Freeman Worth Publishers
Probability Models
In Section 5.1, we used simulation to imitate chance behavior.
Fortunately, we don’t have to always rely on simulations to
determine the probability of a particular outcome.
Descriptions of chance behavior contain two parts:
The sample space S of a chance process is the set of all
possible outcomes.
A probability model is a description of some chance
process that consists of two parts:
• a sample space S and
• a probability for each outcome.
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Ex: Building a probability model: You roll a pair of dice.
Sample Space
36 Outcomes
The Practice of Statistics, 5th Edition
Since the dice are fair, each outcome is equally
likely.
Each outcome has probability 1/36.
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Probability Models
An event is any collection of outcomes from some chance
process. That is, an event is a subset of the sample space.
Events are usually designated by capital letters, like A, B, C,
and so on.
If A is any event, we write its probability as P(A).
In the dice-rolling ex, suppose we define event A as “sum is 5.”
There are 4 outcomes that result in a sum of 5.
Since each outcome has probability 1/36, P(A) = 4/36.
Suppose event B is defined as “sum is not 5.” What is P(B)?
P(B) = 1 – 4/36 = 32/36
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Basic Probability Rules
•For any event A, 0 ≤ P(A) ≤ 1.
•If S is the sample space in a probability model, P(S) = 1.
Meaning all possible outcomes together must have
probabilities whose sum is exactly 1.
•In the case of equally likely outcomes,
number of outcomes corresponding to event A
P(A) =
total number of outcomes in sample space
•Complement rule: The probability an event does not
occur: P(AC) = 1 – P(A)
•Addition rule for mutually exclusive events (disjoint):
If A and B are mutually exclusive (no outcomes in common),
P(A or B) = P(A) + P(B) because P(A and B ) = 0.
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Applying Probability Rules
Distance-learning courses are rapidly gaining popularity among
college students. Randomly select an undergraduate student
who is taking a distance-learning course for credit, and record
the students’ age. Here is the probability model.
Age group (yr): 18 to 23 24 to 29 30 to 39 40 or over
Probability:
0.57
0.17
0.14
0.12
a) Show that this is a legitimate probability model.
a) Find the probability that the chosen student is not in the
traditional college age group (18 to 23 years).
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Two-Way Tables and Probability
When finding probabilities involving two events, a two-way table can
display the sample space in a way that makes probability calculations
easier.
Consider the example on page 309. Suppose we choose a student at
random. Find the probability that the student (a) has pierced ears.
(b) is a male with pierced ears.
(c) is a male or has pierced
ears.
Define events A: is male and B: has pierced ears.
(a)
(b)We
Each
Wewant
want
student
find
isP(male
equally
P(maleorlikely
and
pierced
to be
chosen.
ears),
103
is,
students
P(A
andhave
B).
(c)
totofind
pierced
ears),
thatthat
is, P(A
or B).
There
pierced
ears.
ears)
= P(B)
103/178.
Look
the intersection
the
“Male”
row=and
“Yes”
column.
are
90 at
males
inSo,
the P(pierced
class of
and
103
individuals
with
pierced
ears. There
are 19 males
with pierced
ears. ears
So, P(A
and count
B) = 19/178.
However,
19 males
have pierced
– don’t
them twice!
P(A or B) = (19 + 71 + 84)/178. So, P(A or B) = 174/178
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Check your understanding
A standard deck of playing cards (with jokers removed)
consists of 52 cards in four suits – clubs, diamonds, hearts,
and spades. Each suit has 13 cards, with denominations
ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king. The
jack, queen, and king are referred to as “face cards”.
Imagine that we shuffle the deck and deal one card. Let’s
define events F: getting a face card and H: getting a heart.
1. Make a two-way table that displays the sample space.
2. Find P(F and H)
3. Explain why P(F or H) does not equal P(F) + P(H). Then
use the general addition rule to find P(F or H).
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General Addition Rule for Two Events
Mutually exclusive events have no outcomes in common, what
if the events are not mutually exclusive?
General Addition Rule for Two Events
If A and B are any two events resulting from some chance
process, then P(A or B) = P(A) + P(B) – P(A and B)
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Venn Diagrams and Probability
The complement AC contains exactly the
outcomes that are not in A.
The events A and B are mutually exclusive (disjoint) because
they do not overlap. That is, they have no outcomes in
common.
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Venn Diagrams and Probability
The intersection of events A and B (A ∩ B) is the set of
all outcomes in both events A and B.
The union of events A and B (A ∪ B) is the set of all outcomes
in either event A or B.
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Venn Diagrams and Probability
Recall the example on gender and pierced ears. We can use a Venn
diagram to display the information and determine probabilities.
Define events A: is male and B: has pierced ears.
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