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DU - INFO 1020 - Class Notes - Week 1

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INFO 1020:

Analytics II

Class Notes Mon. 1/04

Chapter 4: Introduction to Probability

• Experiments

• What is an experiment?

• Definition: Any process which has a well-defined set of outcomes

• Example T/F question: “An experiment is a process with a set of known outcomes and a set of unknown outcomes” • Example Experiment: Toss 3 coins and observe Heads or Tails

• Sample Space:

• Definition: Set of all possible outcomes

• Probability

• Definition: A number between and including 0 and 1. It indicates the frequency with which something happens. Can be expressed as a percent, fraction, or decimal

We also discuss several other topics like How do you find the domain of a polynomial graph?

• 3 Methods of Assigning Probability

• Subjective: Personal opinion of the likelihood of something happening

• Empirical: Doing an experiment repeatedly and counting outcomes to determine probability as observed results (rolling dice or tossing coins)

• Classical: Mathematical and logical analysis

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• Events

• What do we mean by “event”?

• Definition: Any subset of the sample space

• Examples: A: exactly 3 heads = 1/8

B: penny is a heads = 4/8

C: nickel is a tails = 4/8

D: more than 1 head = 4/8

E: at most 2 heads = 7/8

• How do I calculate (Classical method) the probability of an event?: Classical probabilities are usually based on counting equally likely outcomes as listed versus

performing an experiment (Empirical)

• Law of Large Numbers: The more you do an experiment, the closer you get to the true probability

• Complement

• Definition: Denoted as A’ and is the event that contains outcomes not in A. P(A) + P(A’) = 1

• Example: P(E) = 7/8 therefore P(E’) = 1 - 7/8 = 1/8 Don't forget about the age old question of How do you explain conservation of energy?

• Union

• Definition: Union of two events, B ∪ C, is the set of outcomes belonging to B OR C OR BOTH

• Example: What is P(B ∪ C)? = 6/8 = P(B) + P(C) - P(B and C) Don't forget about the age old question of What are the 6 basic emotions of feelings?

• Intersection

• Definition: The intersection of two events, B ∩ C, is the set of outcomes belonging to B AND C

• Example: What is P(B ∩ C)? = 2/8

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• What is the Addition Law? We also discuss several other topics like How many ribs do you have altogether in your chest area?

• P(B) + P(C) - P(B and C)

• If events are mutually exclusive, how does the Addition Law change?

• There are no events in common

• How does conditional probability differ from ordinary probability?

• P(A| B) = 1/4: We’ve reduced sample space from 8 to 4. Sample space is usually reduced in cases of conditional probability

• Example: What is P(A|C)? = 0

• Example: What is P(E|D)? = 0

• Frequency Table

MARKETING

MANAGEMENT

ACCOUNTING

BIA

FINANCE

FRESHMAN

4

4

5

1

1

15

SOPHOMORE

2

4

1

6

2

15

JUNIOR

3

6

2

6

3

20

SENIOR

1

1

2

2

4

10

Don't forget about the age old question of What is in the cytoplasm of bacterial cells?

• If events are independent, then P(A | B) = P(A)

• This is how you can prove events are independent

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INFO 1020:

Analytics II

Class Notes Mon. 1/06

Chapter 5a: Discrete Probability Distributions

• Random Variables: Discrete and Continuous

• What is a random variable?

• Definition: “x” whose value is a number which is dependent on the outcome of some experiment or observation

• Example: Roll 2 dice

• R.V. 1: Let x be the distance between the dice

• R.V. 2: Let x be the sum of numbers on the tops If you want to learn more check out How does content play a role in the view of formal analysis?

• R.V. 3: Let x be the product of numbers on tops

• R.V. 4: Let x be the time it takes to stop rolling

• What makes a RV discrete?

• Definition: x-values which are listable (R.V. 2 & R.V. 3). They do not have to be whole numbers.

• What makes a RV continuous?

• Definition: x-values which are un-listable (R.V. 1 & R.V. 4). They do not have to be whole numbers.

• Examples: temperature, age, time, distance

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• Discrete Probability Distributions

• What is a probability function?

• Definition: A formula or a graph which provides all the x values and associated P(x) - values

• formula: P(x) = 1/2x Ɐ x ϵ {1,2,3,6}

• table:

x

P(x)

1

1/2

2

1/4

3

1/6

6

1/12

• graph: x on x-axis and P(x) on y-axis

• Can I list the required conditions for a Discrete Probability Function?

• 3 Requirements:

• 1: Every x-value

• 2: Must have every P(x) - value

• 3: ΣP(x) = 1

• Create a DPF: table and graph

• Experiment: Roll two dice and multiply top two numbers • table

x

P(x)

1

1/36

2

2/36

3

2/36

4

3/36

5

2/36

6

4/36

8

2/36

x

P(x)

9

1/36

10

2/36

12

4/36

15

2/36

16

1/36

18

2/36

20

2/36

x

P(x)

24

2/36

25

1/36

30

2/36

36

1/36

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• graph

• How do I know if a DPF is uniform?

• Definition: P(x) = k Ɐ x

• Example:

x

P(x)

1

1/6

2

1/6

3

1/6

4

1/6

5

1/6

6

1/6

• How do we calculate the expected value of a DRV? • To calculate expected value of x: E(x), the average x, #, = Σx⋅P(x)

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• How do we calculate the VAR and STDEV of X? • To calculate variance (VAR): (x-E(x))² ⋅ P(x)

• To calculate standard deviation: square root of variance Page 4 of 4