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Probability Theory: Interpretations, Rules, and Events, Study notes of Reasoning

An introduction to probability theory, covering different probability interpretations such as theoretical, empirical, and subjective, as well as rules like the Complement, General Addition, and Multiplication rules. It also explains mutually exclusive events and independent events.

What you will learn

  • What are the different probability interpretations?
  • What are mutually exclusive events?
  • What is the General Addition Rule and how is it used?
  • How does the Complement Rule work?
  • How does the Multiplication Rule apply to independent events?

Typology: Study notes

2021/2022

Uploaded on 09/12/2022

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Probability Interpretations
Theoretical. Where P comes from mathematical model or
logical reasoning. (pp. 193-4)
Example: P(‘heads’) = exactly 1/2
Empirical. Where P comes is the long-run relative
frequency of an event. (p. 192)
Example: P(student graduates) = 1/400
Subjective. Where P is a personal degree of belief. (p. 194)
Example: P(I can cross street safely) = 0.99
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Download Probability Theory: Interpretations, Rules, and Events and more Study notes Reasoning in PDF only on Docsity!

Probability Interpretations

  • Theoretical. Where P comes from mathematical model or logical reasoning. (pp. 193-4) - Example: P(‘heads’) = exactly 1/
  • Empirical. Where P comes is the long-run relative frequency of an event. (p. 192) - Example: P(student graduates) = 1/
  • Subjective. Where P is a personal degree of belief. (p. 194)
    • Example: P(I can cross street safely) = 0.

Complement Rule

  • The probability that an event occurs is l minus the probability that it doesn't occur. (p. 195)

P(A) = 1 – P(~A) or P(A) + P(~A) = 1

General Addition Rule

For any two events A and B, the probability of A or B occurring is :

P(A or B) = P(A) + P(B) – P( A and B).

Example: General Addition Rule

Flip a coin twice. Probability of 2 heads:

P(heads, heads) = 0.5 + 0.5 – 0.25 = 0.75.

Example: Mutually Exclusive Events

  • Being registered as Republican vs. Democrat vs. Independent are mutually exclusive events.
  • Freshman, Sophomore, Junior, Senior
  • CA driver’s license lists gender as M or F
  • Rolling a ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, or ‘6’ on fair die

Addition Rule for Mutually

Exclusive Events

If A and B are mutually exclusive events, then the probability of A or B is:

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

p. 196

General Multiplication Rule

  • For any two events (A, B), the probability of A and B is:

P(A and B) = P(A) × P(B|A)

p. 202

Example: General Multiplication Rule

  • Probability of being a Democrat and likes president:

= P(Democrat) × P(likes president|Democrat) = 0.50 × 0.80 = 0.

Example: Independent Events

  • Getting ‘heads’ on one coin clip independent of getting ‘heads’ on another.
  • Rolling a ‘1’ on a fair die independent of getting ‘heads’ on a coin clip.

P(‘1’|heads) = P(‘1’|tails) = P(‘1’)

Multiplication Rule for Independent Events

  • If A and B are independent, the probability of A and B is:

p. 202

P(A and B) = P(A) × P(B)

Conditional Probability

  • The conditional probability P(B|A) is the probability of B given that A occurs.

pp. 202-

P(B|A) =

P(A and B) P(A)

Example: Conditional Probability

Table 1. Purchase Type by Gender

P(male customer|buys utility lighting) = 0.40/0.50 = 0.80.