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 ProbabilityRelative 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 ProbabilityBasic 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 B Venn Diagram

Pr [ A U B ] = Pr [A] +Pr [B]Pr [ A U B ] = Pr [occurrence of A or B or Both]

Basic ProbabilityBasic 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 BA∩B

Pr [ A U B ] = Pr [occurrence of A or B or Both]

Pr [ A U B ] = Pr [A] +Pr [B] – Pr [A ∩ B]

Pr [A ∩ B] = Pr [occurrence of both A and B]

Venn Diagram

Basic ProbabilityBasic 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 ProbabilityBasic 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 to give the probability of them occurring together.

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 ProbabilityBasic 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 ProbabilitySummary 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.