Transcript Basic Probability

Basic Probability
(Chapter 2, W.J.Decoursey, 2003)
Objectives:
- Define probability and its relationship to relative
frequency of an event.
- Learn the basic rules of combining probabilities.
- Understand the concepts of mutually exclusive / not
mutually exclusive and independent / not
independent events.
- Apply these concepts to solve sample problems.
Basic Probability
Probability:
A measure of the likelihood that a particular
event will occur.
e.g. If we are certain that an event will occur, its
probability is 1 or 100%.
If it certainly will not occur, its probability is
zero.
What is the probability of rolling a one using an
eight sided dice.
Basic Probability
Relative Frequency:
An estimate of the true probability of an event. For a large
number of trials, we get a very good estimate of that
probability. For an infinite number of trials, they would be
identical.
e.g. 260 bolts are examined as they are produced. Five of
them are found to be defective. On the basis of this
information, estimate the probability that a bolt will be
defective.
Answer: The probability of a defective bolt is approximately
equal to the relative frequency, which is
5/260=0.019=1.9%
Basic Probability
Basic Rules of Combining Probability:
A final outcome is normally arrived at through a
series of events. Each event has its own
probability of occurring. We must combine these
events in an appropriate manner to determine
the probability of the final outcome.
- Addition Rule
- Multiplication Rule
Basic Probability
Basic Rules of Combining Probability:
- Addition Rule:
Pr [event A] = C, Pr [event B] = D, what is the probability of
Pr [event A or event B] ? Equals C+D?
Case one: Mutually exclusive events: if one event occurs,
other events can not occur.
The
probability of occurrence of one or another of more than
one event is the
of the probabilities of the separate
events.
Example one.
A
Pr [ A U B ] = Pr [A] +Pr [B]
Pr [ A U B ] = Pr [occurrence of A or B or Both]
B
Venn Diagram
Basic Probability
Basic Rules of Combining Probability:
- Addition Rule:
Case two: Not mutually exclusive events: there can be
overlap between them. The probability of overlap must
be subtracted from the sum of probabilities of the
separate events.
Example 2
A
A∩B
B
Pr [ A U B ] = Pr [A] +Pr [B] – Pr [A ∩ B]
Pr [ A U B ] = Pr [occurrence of A or B or Both]
Pr [A ∩ B] = Pr [occurrence of both A and B]
Venn Diagram
Basic Probability
Basic Rules of Combining Probability:
- Multiplication Rule:
a. The basic idea for calculating the number of choices:
- There are n1 possible results from one operation.
- For each of these, there are n2 possible results from a
second operation.
- Then here are (n1Xn2) possible outcomes of the two
operations together.
e.g. In one case a byte is defined as a sequence of 8 bits.
Each bit can be either zero or one. How many different
bytes are possible?
Solution: we have 2 choices for each bit and a sequence of
8 bits. Then the number of possible results is 28=256.
Basic Probability
Basic Rules of Combining Probability:
- Multiplication Rule:
b. Independent events
The occurrence of one event does not affect the
probability of the occurrence of another event.
The probability of the individual events
give the probability of them occurring together.
to
Consider 2 events, A and B. Then the probability of
occurring together is
Pr [A] * Pr [B]
Note the use of logical
Example 3
Basic Probability
Basic Rules of Combining Probability:
- Multiplication Rule:
c. Not-independent events
The occurrence of one event affects the probability of the
occurrence of another event.
The probability of the affected event is called the conditional
probability since it is conditional upon the first event
taking place.
The multiplication rule then becomes for
occurring
together
Pr [A ∩ B]=Pr [A] * Pr [B|A]
Pr [B|A] : conditional probability of B. (examples in class)
Basic Probability
Summary of Combining Rule
When doing combined probability problems, ask yourself:
1. Does the problem ask the logical OR or the logical
AND?
2. If OR, ask your self are the events mutually exclusive
or not? If yes, Pr [ A U B ] = Pr [A] +Pr [B],
other wise
Pr [ A U B ] = Pr [A] +Pr [B] – Pr [A ∩ B]
3. If AND, use the multiplication rule and remember
conditional probability. A probability tree may be helpful.