Chapter
6: Probability

6.1 The Idea of Probability (pp. 330-333)

1. In
statistics, what is meant by the term *random*? We call phenomenon __random__
if individual outcomes are uncertain but there in nonetheless a regular
distribution of outcomes in a large number of repetitions. “Random” in statistics is not a synonym for
“haphazard” but a description of a kind of order that emerges only in the long
run.

2. In
statistics, what is meant by *probability*? The __probability__
of any outcome of a random phenomenon is the proportion of times the outcome
would occur in a very long series of repetitions. That is, probability is a long-term relative
frequency.

3. What
is *probability theory*? __Probability
theory__ is the branch of mathematics that
describes random behavior.

4. In statistics, what is meant by an *independent* trial? Trials are __independent__
if the outcome

of one trial does not influence the
outcome of any other.

6.2 Probability Models (pp.
335-356)

1. In statistics, what is a *sample space*? The __sample space
S__ of a random phenomena is the set

of all possible outcomes.

2. In statistics, what is an *event*?
An __event__ is any outcome or a set of
outcomes of a random

phenomenon. That is, an event is subset of the sample
space.

3. What is a probability model? A __probability
model__ is a mathematical description of a random

phenomenon consisting of two
parts: a sample space S and a way of
assigning probabilities

to events.

4. When counting the number of events in a
sample space, what is meant by the *multiplication
principle? *If you can do one
task in

ways, then both tasks can be done in

5. What is the difference between sampling with
replacement and sampling without

replacement? For example, if you
are selecting random digits by drawing numbered slips of

paper from a hat, and you want all
ten digits to be equally likely to be selected each draw,

then after you draw a digit and
record it, you must put it back into the hat.
Then the second

draw will be exactly like the
first. This is referred to as sampling __with
replacement__. If you

do not replace the slips you draw,
however, there are only nine choices for the second slip

picked, and eight for the
third. This is called sampling __without
replacement__.

6. Explain why the probability of any *event* is a number between 0 and 1. An __event__ with

probability 0 never occurs. An event with probability 1 occurs on every
trial. The

probability of any event must be
between these two possible outcomes.

7. What is the sum of the probabilities of all
possible *outcomes* in a sample space? Because

some outcome must occur on every
trial, the sum of the probabilities for all possible

outcomes must be exactly 1.

8. Describe the probability that an *event* does not occur. The probability that
an event does not

occur is 1 minus the probability
that the event does occur. The probability
that an event

occurs and the probability that it
does not occur always add to 100% or 1.

9. What is meant by the *complement* of an event? The __complement__ of any event *A* is the event that *A*
does not occur, written as _{}. The complement rule states that _{}.

10. When are two events considered *disjoint*? Two events *A* and *B* are __disjoint__ (also called

__mutually exclusive__) if they
have no outcomes in common and so can never occur

simultaneously.

11. What is the probability of two *disjoint* events? If *A* and *B* are disjoint. Then

*P(A or B) = P(A) + P(B).*
This is the __addition rule__ for disjoint events.

12. How do you find the probability of equally
likely events in a sample space? If a random

phenomenon has *k* possible outcomes, all equally likely,
then each individual outcome has

probability 1/*k*.
The probability of any event *A*
is

_{}

13 .Explain why the probability of getting
heads when flipping a coin is 50%. P(heads)_{}

14. What is the *Multiplication Rule* for *independent* events? Two events *A* and *B* are

independent if knowing that one
occurs does not change the probability that the other

occurs. If *A*
and *B* are independent, *P(A and B) = P(A)P(B).*

15. Can *disjoint*
events be *independent*? No, disjoint events
cannot be __independent__. If *A* and *B*

are disjoint, then the fact that *A* occurs tells us that *B* cannot occur.

16. If two events *A* and *B* are *independent*, what must be true about *A ^{c}* and

independent then

6.3
General Probability Rules
(pp. 359-379)

1. What
is meant by the *union* of two or more
events? Draw a diagram. The __union__ of
any

collection of events is the event
that at least one of the collection occurs.

2.
State the addition rule for *disjoint*
events. If
events *A*, *B*, and *C* are __disjoint__
in the sense that

no two have any outcomes in
common, then *P(one or more of A, B, C) = P(A) + P(B) + P(C).*

3. State the general addition rule for *unions* of two events. For any two events *A* and *B*,

*P(A or B) = P(A) + P(B) – P(A and B).*

4. Explain the difference between the rules in
#2 and #3. If *A* and *B* are disjoint, the event

{*A* and *B*} that both occur has not outcomes in
it. This empty event _{} is the complement of
the sample space S and must have probability 0.
So the general addition rule #3 includes #2, the addition rule for
disjoint events.

5. What is meant by *joint probability*? The simultaneous occurrence of two events is called a

joint event. The probability of a joint event is called a __joint
probability__.

6. What is meant by *conditional probability*? The conditional probability *P(B|A)* is the

probability that event *B* occurs given that event *A* has already occurred.

7. State the __general multiplication rule__. The probability that both of two events *A* and *B* happen

together can be found by *P(A and B) = P(A)P(B|A)*

8. How is the general multiplication rule
different than the multiplication rule for independent

events? If *A* and *B* are independent, then *P(B|A)
= P(B)* so the two rules are the same.
If *A*

and *B* are *not* independent,
the rule must be adjusted for this condition.

9. State the formula for finding conditional
probability. When
*P(A)* > 0, the conditional

probability of *B* given *A* is: *P(B|A)*
_{}.

10. What is meant by the *intersection* of two or more events? Draw a diagram. The __intersection__

of any collection of events is the
event that all of the events occur.

11. Explain the difference between
the *union* and the *intersection* of two or more events. The

union is the event that __any__
of them occur, the intersection is the event that they __all__ occur.

*independent*. Two events *A* and *B* that

both have positive probability are
__independent__ if *P(B|A) = P(B).*