s1 w3

• Lecture 3 Topics


  • Review and Preview
  • Basic Concepts of Probability
  • Addition Rule
  • Multiplication Rule: Basics
  • Multiplication Rule: Complements and Conditional Probability
  • Counting
  • Probabilities Through Simulations
  • Bayes' Theorem
• Necessity of sound sampling methods
• Common measures of characteristics of data, such as the mean and the standard deviation
• Rare Event Rule for Inferential Statistics: If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct

  
• We use the rare event rule for inferential statistics
• 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


  For a procedure consists of all possible simple events; that is, the sample space consists of all outcomes that cannot be broken down any further
• Procedure


  • Single birth
  • 3 births
• Example of Event


  • 1 girl (simple event)
  • 2 boys and 1 girl
• Sample Space


  • {b, g}
  • {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}
• Probability Notation


  P - denotes a probability

  A, B, and C - denote specific events

  P(A) - denotes the probability of event A occurring

  
• Relative Frequency Approximation of Probability


  P(A) = (# of times A occurred)/(# of times procedure was repeated)
• Classical Approach to Probability (Requires Equally Likely Outcomes)

  P(A) = (number of ways A can occur)/(number of different simple events)
• Subjective Probabilities


  P(A), the probability of event A, is estimated by using knowledge of the relevant circumstances
• As a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability
• Example


  • When three children are born, the sample space is: {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}

  Assuming that boys and girls are equally likely, the probability of getting three children of the same gender is 2/8 = 0.25
• Simulation of a procedure


  A process that behaves in the same ways as the procedure itself so that similar results are produced
• Always express a probability as a fraction or decimal number between 0 and 1

  The probability of an impossible event is 0

  The probability of an event that is certain to occur is 1

  For any event A, the probability of A is between 0 and 1 inclusive
• Unlikely event


  An event with a probability of 0.05 or less
• Unusual outcome


  An event with an unusually low or high number of outcomes of a particular type, far from what is typically expected
• Complement of event A (Ā)

  Consists of all outcomes in which the event A does not occur
• Example


  • 1010 adults were surveyed and 202 of them were smokers

  P(smoker) = 202/1010 = 0.200

  P(not a smoker) = 1 - 202/1010 = 0.800
• When expressing the value of a probability, either give the exact fraction or decimal or round off final decimal results to three significant digits
• Suggestion: When a probability is not a simple fraction, express it as a decimal so that the number can be better understood
• Actual odds against event A


  The ratio P(Ā)/P(A), usually expressed in the form of a:b (or "a to b"), where a and b are integers having no common factors
• Actual odds in favour of event A

  The ratio P(A)/P(Ā), which is the reciprocal of the actual odds against the event
• Payoff odds against event A

  The ratio of the net profit (if you win) to the amount bet
• Example


  • Betting $5 on the number 1 in roulette, with a probability of winning of 1/38 and payoff odds of 35:1

  Actual odds against 1 = (37/38)/(1/38) = 37:1

  Net profit if you win = $35 per $1 bet, so for a $5 bet the net profit is $175
• Lecture 3 Topics


  • Review and Preview
  • Basic Concepts of Probability
  • Addition Rule
  • Multiplication Rule: Basics
  • Multiplication Rule: Complements and Conditional Probability
  • Counting
  • Probabilities Through Simulations
  • Bayes' Theorem
• Addition Rule


  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
• Disjoint (or mutually exclusive) events

  Events A and B that cannot occur at the same time
• Rule of Complementary Events

  P(A) + P(Ā) = 1
• Multiplication Rule: Basics


  Used for finding P(A and B), the probability that event A occurs in a first trial and event B occurs in a second trial
• P(B|A)


  The probability of event B occurring after event A has already occurred
• Formal Multiplication Rule


  P(A and B) = P(A) * P(B|A)
• Intuitive Multiplication Rule


  When finding the probability that event A occurs in one trial and event B occurs in the next trial, multiply the probability of event A by the probability of event B, but be sure that the probability of event B reflects the occurrence of event A
• Multiplication Rule: Complements and Conditional Probability


  Used for finding P(A and B), the probability that event A occurs in a first trial and event B occurs in a second trial
• Multiplication Rule


  1. P(A and B) = P(A) * P(B|A)

  2. When finding the probability that event A occurs in one trial and event B occurs in the next trial, multiply the probability of event A by the probability of event B, but be sure that the probability of event B takes into account the previous occurrence of event A
• When applying the multiplication rule, always consider whether the events are independent or dependent, and adjust the calculations accordingly