Probabilistic & Statistical Techniques Eng. Tamer Eshtawi First Semester Eng. Tamer Eshtawi First Semester

Lecture 6 Chapter 3 (part 1) Probability Main Reference: Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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

Overview Probabilities is a mathematical measure of the likelihood of an event occurring. Probabilities are always fractions or decimals indicating the portion or percent of the time that the event occurs

Section 4-2 Fundamentals

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.

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 the sample space consists of all outcomes that cannot be broken down any further

Example sample spaceThe sample space of the experiment Flip a Coin has 2 outcomes heads and tails, so we could write: sample spaceThe sample space of the experiment of rolling a die has 6 possible outcomes 1-6, so: The Event The Event : Rolling an even number is an event for the experiment of rolling a die. If we call this event E, we could write:

Notation for Probabilities P - denotes a probability. A, B, and C - denote specific events. P (A) - denotes the probability of event A occurring.

Basic Rules for Computing Probability Rule 1: Relative Frequency Approximation of Probability Observe results, 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

Rule 2: Probability for 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 P(A) = number of ways A can occur number of different simple events s n =

Rule 3: Subjective Probabilities P(A), the probability of event A, is estimated by using knowledge of the relevant circumstances.

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.

Probability Limits  The probability of an event that is certain to occur is 1.  The probability of an impossible event is 0.  For any event A, the probability of A is between 0 and 1 inclusive. That is, 0  P(A)  1.

Example 1 In a class, 18 students own computers and 7 do not. If one of the student is randomly selected, find the probability of getting one who does not own a computer

Example 2 Find the probability that a couple with 3 children will have exactly 2 boys. The gender of any child is not influenced by the gender

Example 3 When two balanced dice are rolled, 36 equally likely outcomes are possible: a) find The probability the sum is 11, b) the two dice are doubles The sum of the dice can be 11 in two ways. The probability the sum is 11 = 2/36 = Doubles can be rolled in six ways. The probability of doubles = 6/36 =

Possible Values for Probabilities

Definition The complement of event A, denoted by, consists of all outcomes in which the event A does not occur.

Example 4 The General motor co. wanted to test a new model. 50 drivers has been recruited, 20 of whom are men. When the first person is selected, what the probability of not getting a male driver?

Recap In this section we have discussed:  Rare event rule for inferential statistics.  Probability rules.  Law of large numbers.  Complementary events.

Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Section 4-3 Addition Rule

Key Concept The main objective of this section is to present the addition rule as a device for finding probabilities that can be expressed as P(A or B), the probability that either event A occurs or event B occurs (or they both occur) as the single outcome of the procedure.

Compound Event Any event combining 2 or more simple events Definition Notation P(A or B) = P (in a single trial, event A occurs or event B occurs or they both occur)

When finding the probability that event A occurs or event B occurs, find the total number of ways A can occur and the number of ways B can occur, but find the total in such a way that no outcome is counted more than once. General Rule for a Compound Event

Compound Event Formal Addition Rule P(A or B) = P(A) + P(B) – P(A & B) where P(A and B) denotes the probability that A and B both occur at the same time as an outcome in a trial or procedure.

Definition Events A and B are disjoint if they cannot occur at the same time. (That is, disjoint events do not overlap.) Events That Are Not DisjointDisjoint Events

Complementary Events P(A) and P(A) are disjoint It is impossible for an event and its complement to occur at the same time.

Rules of Complementary Events

Venn Diagram for the Complement of Event A If P(A) = 0.3 P(A) = 1 – P(A) = 1 – 0.3 = 0.7

Example 5 Titanic Passengers (Table 3-1), Assuming that 1 person is randomly selected from 2223 people abroad the titanic: Find P(selected a man or a boy) Find P(selected a man or some one who survived)

Solution P(selected a man or a boy) = P (men) + P (boys) - 0 = P(selected a man or survived) = P (men) + P (survived) – P(men & survived) =

Recap In this section we have discussed:  Compound events.  Formal addition rule.  Disjoint events.  Complementary events.

Flash points

A bag contains 6 red marbles, 3 blue marbles, and 7 green marbles. If a marble is randomly selected from the bag, what is the probability that it is blue? A. 1/3 B. 1/7 C. 3/16 D. 1/13

A bag contains 8 red marbles, 4 blue marbles, and 1 green marble. Find P(not blue). A. 9/13 B. 9 C. 13/9 D. 4/13