Introduction to Probability (Dr. Monticino)

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Introduction to Probability
(Dr. Monticino)
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Assignment Sheet

• Read Chapters 13 and 14
• Assignment 8 (Due Wednesday March 23rd )
• Chapter 13
• Exercise Set A 1-5 Exercise Set B 1-3
• Exercise Set C 1-4,7 Exercise Set D 1,3,4
• Review Exercises 2,3, 4,5,7,8,9,11
• Chapter 14
• Exercise Set A 1-4 Exercise Set B 1-4, 5
• Exercise Set C 1,3,4,5 Exercise Set D 1
  (just calculate probabilities)
• Review Exercises 3,4,7,8,9,12

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Overview

• Framework
• Equally likely outcomes
• Some rules

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Probability Framework

• The sample space, ?, is the set of all outcomes
  from an experiment
• A probability measure assigns a number to each
  subset (event) of the sample space, such that
• 0 ? P(A) ? 1
• P(? ) 1
• If A and B are mutually exclusive (disjoint)
  subsets, then P(A ? B) P(A) P(B) (addition
  rule)

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Equally Likely Outcomes

• Outcomes from an experiment are said to be
  equally likely if they all have the same
  probability.
• If there are n outcomes in the experiment then
  the outcomes being equally likely means that each
  outcome has probability 1/n
• If there are k outcomes in an event, then the
  event has probability k/n
• Fair is often used synonymously for equally
  likely

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Examples

• Roll a fair die
• Probability of a 5 coming up
• Probability of an even number coming up
• Probability of an even number or a 5
• Roll two fair die
• Probability both come up 1 (double ace)
• Probability of a sum of 7
• Probability of a sum of 7 or 11

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More Examples

• Spin a roulette wheel once
• Probability of 11
• Probability of red probability of black
  probability of not winning if bet on red
• Draw one card from a well-shuffled deck of cards
• Probability of drawing a king
• Probability of drawing heart
• Probability of drawing king of hearts

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Conditional Probability

• All probabilities are conditional
• They are conditioned based on the information
  available about the experiment
• Conditional probability provides a formal way for
  conditioning probabilities based on new
  information
• P(A B) P(A ? B)/P(B)
• P(A ? B) P(A B) ? P(B)

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Multiplication Rule

• The probability of the intersection of two events
  equals the probability of the first multiplied by
  the probability of the second given that the
  first event has happened
• P(A ? B) P(A B) ? P(B)

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Examples

• Suppose an urn contains 5 red marbles and 8 green
  marbles
• Probability of red on first draw
• Red on second, given red on first (no
  replacement)
• Red on first and second

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Independence

• Intuitively, two events are independent if
  information that one occurred does not affect the
  probability that the other occurred
• More formally, A and B are independent if
• P(A B) P(A)
• P(B A) P(B)
• P(A?B) P(A)P(B)
• 
  (Dr. Monticino)