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We should appreciate, cherish and cultivate blessings.

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Chapters 10, 12: Probability

Sample space/eventProbability modelsBasic probability rulesRandom variables

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Random Phenomenon & Probability

A random phenomenon is one in which the outcome is unpredictable. The outcome is unknown until we observe it

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

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Example 1: Traffic Jam on the Golden Bridge

< = 20 mph traffic jam How often will a driver encounter traffic jam

on the golden bridge during 7-9 am weekdays? __ out of 100 times.

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Example 2: Coin Flipping

What is the probability that a flipped coin shows heads up?

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Two Ways to Determine Probability

Making an assumption about the physical world and use it to find proportions

Observe outcomes and find its proportions in a very long series of repetitions

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Probability Models

Sample space: the collection of all possible outcomes of a random phenomenon

An event is an outcome or a set of outcomes of a random phenomenon; i.e. a subset of the sample space

A probability model consists of two parts: a sample space S and a way of assigning probabilities to events; i.e. a model for distribution

** what are the sample space/event of the examples?

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Rules for a Probability Model

The probability P(A) of any event A is between 0 and 1

The probability P(S) of sample space is 1

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Building a Probability Model

(Theoretical way) Making an assumption about the physical world and use it to build the model

(Data way) Measuring a representative sample and observing proportion of the sample that fall into various outcomes

Do example 2 and then example 3

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Example 3: Employment

What is the probability that a CSUEB graduate gets hired within 3 months after graduation? Between 3 to 6 months? More than 6 months?

** Data: In a representative sample of 200 graduates, 150, 30, 20 of them got hired within 3, 3-6, and > 6 months, respectively.

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Probability of an Event

The sum of the probabilities of outcomes in the event

** Revisit the examples 2, 3 and find the events & their probabilities

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Mutually Exclusive Events

Two events are mutually exclusive if they do not contain any of the same outcomes. They are also called disjoint.

Basic Probability Rules (p. 269)

Rule 3: If two events A and B are disjoint, then

P(A or B) = P(A)+P(B)

Rule 4: For any event A,

P(A does not occur) = 1- P(A)

** The event (A or B) occurs if either A or B or both occur.

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Probability Models

A probability model with a finite sample space is called discrete

Example: problem 10.11 (p. 272)

A continuous probability model assigns probabilities as areas under a density curve. The area under the curve and above any range of values is the probability of an outcome in that range

Example: randomly pick a number between 0 and 1

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Random Variables

A random variable is a variable whose value is a numerical outcome of a random phenomenon

The distribution of a random variable X tells us what values X can take and how to assign probabilities to those values

Example: 1. # of dots (problem 10.11) and

2. height of a young lady (example 10.9)

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Independent Events

Two events are independent if the probability that one event occurs stays the same, no matter whether or not the other event occurs.

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Conditional Probability

The conditional probability of the event B given the event A, denoted P(B|A), is the long-run relative frequency with which event B occurs when circumstances are such that event A also occurs.

** event A = age 21+; event B = female

** Ask students for age and find P(B|A)

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Basic Rules for Finding Probabilities

P(A or B) = P(A) + P(B) - P(A and B) k disjoint events:

P(A1 or A2 or … or Ak) = P(A1) + P(A2) + … + P(Ak)

P(A and B) = P(A)P(B|A) k independent events:

P(A1 and A2 and … and Ak) = P(A1)P(A2)…P(Ak)

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Steps for Finding Probabilities

1. Identify random phenomenon2. Identify the sample space3. Build the probability model as much as you can4. Specify the event for which the probability is

wanted5. Use the probability model from step 3 and the

probability rules to find the probability of interest

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Tree diagrams

For a sequence of events, when conditional probabilities for events based on previous events are known

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Example:

People are classified into 8 types. For instance, Type 1 is “Rationalist” and applies to 15% of men and 8% of women. Type 2 is “Teacher” and applies to 12% of men and 14% of women. Each person fits one and only one type.

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•What is the probability that a randomly selected male is “Rationalist”? “Teacher”? Both?

•What is the probability that a randomly selected female is not a “Teacher”?

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Suppose college roommates have a particularly hard time getting along with each other if they are both “Rationalists.”A college randomly assigns roommates of the same sex.

What proportion of male roommate pairs will have this problem?

What proportion of female roommate pairs will have this problem?

Assuming that half of college roommate pairs are male and half are female. What proportion of all roommate pairs will have this problem?