Chapter 4 Basic Probability and Probability Distributions
Probability Terminology • Classical Interpretation : Notion of probability based on equal likelihood of individual possibilities (coin toss has 1/2 chance of Heads, card draw has 4/52 chance of an Ace). Origins in games of chance. – Outcome : Distinct result of random process ( N = # outcomes) – Event : Collection of outcomes ( N e = # of outcomes in event) – Probability of event E : P (event E ) = N e / N • Relative Frequency Interpretation : If an experiment were conducted repeatedly, what fraction of time would event of interest occur (based on empirical observation) • Subjective Interpretation : Personal view (possibly based on external info) of how likely a one-shot experiment will end in event of interest
Obtaining Event Probabilities • Classical Approach – List all N possible outcomes of experiment – List all N e outcomes corresponding to event of interest ( E ) – P (event E ) = N e / N • Relative Frequency Approach – Define event of interest – Conduct experiment repeatedly (often using computer) – Measure the fraction of time event E occurs • Subjective Approach – Obtain as much information on process as possible – Consider different outcomes and their likelihood – When possible, monitor your skill (e.g. stocks, weather)
Basic Probability and Rules • A,B ≡ Events of interest • P ( A ), P ( B ) ≡ Event probabilities • Union : Event either A or B occurs ( A ∪ B ) • Mutually Exclusive : A, B cannot occur at same time – If A,B are mutually exclusive: P (either A or B ) = P ( A ) + P ( B ) • Complement of A : Event that A does not occur ( Ā ) – P ( Ā ) = 1- P ( A ) That is: P ( A ) + P ( Ā ) = 1 • Intersection : Event both A and B occur ( A ∩ B or AB ) • P ( A ∪ B ) = P ( A ) + P ( B ) - P ( AB )
Conditional Probability and Independence • Unconditional/Marginal Probability : Frequency which event occurs in general (given no additional info). P ( A ) • Conditional Probability : Probability an event ( A ) occurs given knowledge another event ( B ) has occurred. P ( A | B ) • Independent Events : Events whose unconditional and conditional (given the other) probabilities are the same ( ) ( ) ( | ) ( ) ( ) ( ) ( ) ( | ) ( ) ( ) ( ) ( ) ( ) ( | ) ( ) ( | ) , independent ( ) ( | ) & ( ) ( | ) P A B P AB P A B P B P B P A B P AB P B A P A P A P A B P AB P A P B A P B P A B A B P A P A B P B P B A ∩ = = ∩ = = ∩ = = = ⇒ = =
John Snow London Cholera Death Study • 2 Water Companies (Let D be the event of death): – Southwark&Vauxhall ( S ): 264913 customers, 3702 deaths – Lambeth ( L ): 171363 customers, 407 deaths – Overall: 436276 customers, 4109 deaths people) 10000 per (24 0024 . 171363 407 ) | ( people) 10000 per (140 0140 . 264913 3702 ) | ( people) 10000 per (94 0094 . 436276 4109 ) ( = = = = = = L D P S D P D P Note that probability of death is almost 6 times higher for S&V customers than Lambeth customers (was important in showing how cholera spread)
John Snow London Cholera Death Study Cholera Death Water Company Yes No Total S&V 3702 (.0085) 261211 (.5987) 264913 (.6072) Lambeth 407 (.0009) 170956 (.3919) 171363 (.3928) Total 4109 (.0094) 432167 (.9906) 436276 (1.0000) Contingency Table with joint probabilities (in body of table) and marginal probabilities (on edge of table)
John Snow London Cholera Death Study WaterUser S&V L .6072 .3928 Company Death D (.0085) .0140 .9860 D C (.5987) .0024 .9976 D (.0009) D C (.3919) Tree Diagram obtaining joint probabilities by multiplication rule
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