Events. Independence. Coin Tossing. Random Phenomena


 Lillian Davidson
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1 Random Phenomena Events A random phenomenon is a situation in which we know what outcomes could happen, but we don t know which particular outcome did or will happen For any random phenomenon, each attempt, or trial, generates an outcome Something happens on each trial, and we call whatever happens the outcome The Sample Space S set of all possible outcomes of a random phenomena An event consists of a combination of outcomes Subsets of the sample space The probability of an event is its longrun relative frequency Slide 141 Slide 142 Coin Tossing Independence Relative Frequency In order to think about what happens with combinations of outcomes, it really simplifies things if the individual trials are independent Roughly speaking, this means that the outcome of one trial doesn t influence or change the outcome of another No of Tosses Slide 143 Slide 144
2 The Law of Large Numbers Probability The Law of Large Numbers (LLN) says that the longrun relative frequency of repeated independent events gets closer and closer to the true relative frequency as the number of trials increases The idea of the long run is hard to grasp, so the LLN is often misunderstood The common (mis)understanding is that random phenomena are supposed to compensate some for whatever happened in the past This is just not true Thanks to the LLN, we know that relative frequencies settle down in the long run, so we can officially give the name probability to that value Probabilities must be between 0 and 1, inclusive A probability of 0 indicates impossibility A probability of 1 indicates certainty Slide 145 Slide 146 Examples of Sample Spaces Rolling Two Dice Sample Space Toss a coin twice S = { HH, HT, TH, TT } Roll a pair of die and record numbers S = {(1,1),(1,2),,(1,6),(2,1),, (2,6),,(6,6)} Roll a pair of die and record total score S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} Toss a coin until first tail appears S = {T, HT, HHT HHHT, } Measure duration of charge of mobile phone battery S = { t t 0 } First Die Second Die Slide 147 Slide 148
3 Events, Combining Events Example: Rolling a die Events uppercase letters, A,, C, Special events: S = certain event = null, impossible event = { } Complement of event A is Ā = all outcomes not in A A Union of events; A or or both A Intersection of events A and Disjoint events cannot occur together, ie A = A = score on die is even = { } = score on die is odd = { } C = score is greater than 4 = { } A = A = A C = C = (A C) ( C) = (A ) C = Slide 149 Slide Example: Rolling two dice Probability Distributions A = score on 1 st die is even = { } = score on 2 nd die is odd = { } C = score is greater than 9 = { } A = A = A C = C = (A C) ( C) = (A ) C = Sample space S = {s 1, s 2, s 3, } Probabilities: numbers p 1, p 2, p 3, All p i s lie between 0 and 1 ( 0 p i 1 ) Sum of all p i s is 1: ( p 1 + p 2 + p 3 + = 1 ) Probability of an event obtained by adding up probabilities of all outcomes in A Slide Slide 1412
4 Equally Likely outcomes Formal Probability Example: Draw a card from a well shuffled pack A = event of drawing an Ace 4 P(A) = Generally P(A) = Number of outcomes in A Number of outcomes in S 1 Two requirements for a probability: A probability is a number between 0 and 1 For any event A, 0 P(A) 1 2 Something has to happen rule : The probability of the set of all possible outcomes of a trial must be 1 P(S) = 1 (S represents the set of all possible outcomes) Slide Slide Formal Probability (cont) Formal Probability (cont) 3 Complement Rule: Definition: The set of outcomes that are not in the event A is called the complement of A, denoted A C, or Ā The probability of an event occurring is 1 minus the probability that it doesn t occur P(A) = 1 P(A C ) 4 Addition Rule: Definition: Events that have no outcomes in common (and, thus, cannot occur together) are called disjoint For two disjoint events A and, the probability that one or the other occurs is the sum of the probabilities of the two events P(A ) = P(A or ) = P(A) + P(), provided that A and are disjoint Slide Slide 1416
5 Formal Probability (cont) 5 Multiplication Rule: For two independent events A and, the probability that both A and occur is the product of the probabilities of the two events P(A and ) = P(A ) = P(A) x P(), provided that A and are independent Draw a card: event A an Ace; event a heart 1 A Ā Slide Slide What Can Go Wrong? A Ā P(A ) P(Ā ) P() P(A ) P(Ā ) P() P(A) P(Ā) 100 eware of probabilities that don t add up to 1 Don t add probabilities of events if they re not disjoint Don t multiply probabilities of events if they re not independent Don t confuse disjoint and independent disjoint events can t be independent Slide Slide 1420
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