2. A SMIDGEON ABOUT PROBABILITY AND EVENTS. Wisdom ofttimes consists of knowing what to do next. Herbert Hoover

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1 CIVL 303 pproximation and Uncertainty JW Hurley, RW Meier MIDGEON BOUT ROBBILITY ND EVENT Wisdom ofttimes consists of knowing what to do next Herbert Hoover DEFINITION Experiment any action or process that generates observations (eg flipping a coin) Trial a single instance of an experiment (one flip of the coin) Outcome the observation resulting from a trial ( heads ) ample pace the set of all possible outcomes of an experiment ( heads or tails ) Event a collection of one or more outcomes that share some common trait Mutually Exclusive Events events (sets) that have no outcomes in common Independent Events events whose probability of occurrence are unrelated VENN DIGRM Venn diagrams are graphical representations of events rectangle is used to represent the sample space and closed curves are used to represent events (sets) inside the sample space: For example, using Venn diagrams, two mutually exclusive events could be illustrated as: B

2 ET THEORY Let represent one set of outcomes and B represent another set of outcomes (as illustrated in the Venn diagram on the previous page) The complement of (written ) is the set consisting of all outcomes in the sample space that are not contained in set (ie, not ) The union of and B (written B ) is the set consisting of all outcomes that are either in or in B or in both sets Using a Venn diagram, how would you illustrate that? B The intersection of and B (written B ) is the set consisting of all outcomes that are in both set and set B at the same time How would you illustrate that? B We ll refer back to these diagrams throughout the next section as we discuss the basic laws and axioms of probability 6

3 DEFINITION OF ROBBILITY When conducting an experiment, the probability of obtaining a specific outcome can be defined from its relative frequency of occurrence: () = lim n n N For example, the probability of a coin flip coming up heads could be found by flipping a coin over and over again, keeping track of the number of times you flip the coin (N) and the number of times it comes up heads (n) t first, the odds may not be apparent because the coin flip may come up heads three or four times in a row, then come up tails three or four times in a row, and so on s the number of coin flips gets large, however, the ratio of heads to flips should settle down to ½ (if the coin is not rigged!) BIC XIOM OF ROBBILITY For any event, the probability of occurrence of is always greater than or equal to zero: () 0 By definition, the probability of occurrence of the sample space,, is equal to one: () = This may not be immediately obvious, but every experiment has an outcome and the sample space contains all possible outcomes, so the probability that the outcome of an experiment will be contained in the sample space is 00% Now, recalling our earlier Venn diagrams, if is a set in sample space, the complement of is everything in the sample space not included in Therefore, we can calculate the probability of not as ( ) ( ) = ( ) = ( ) Negation In general, the probability of occurrence of the union of two sets can be written as ( B) = ( ) + B ( ) ( B) Two sets are said to be mutually exclusive if they have no points in common In that case, ( B ) = 0 and the probability of occurrence reduces to ( B) = ( ) + B ( ) The ddition Rule 7

4 CONDITIONL ROBBILITY Conditional probability is the probability of occurring given that B has already occurred In other words, the probability of occurring is conditioned on whether or not B has occurred By definition ( B) = ( B) ( B) Why? If B has already occurred, then the sample space is no longer it s B! That means the only possible outcomes in set are those formerly occupying the intersection of and B o the probability of given B has to be defined relative to the probability of B having already occurred If we rearrange the equation above, we get an expression for the probability of occurrence of the intersection of two sets: ( ) ( ) ( B) = B B Two events and B are said to be independent if the occurrence of is not conditioned on whether or not B has already occurred Therefore, for independent events ( ) = ( ) B and the probability of both and B occurring is ( B) = ( ) B ( ) The Multiplication Rule o, two events, and B, are independent events if and only if the probability of their both occurring is equal to the product of their individual probabilities of occurrence Otherwise, the probabilities of and B depend on each other whether or not there is a causal explanation for that dependence For example, let s say the miths are ardent tennis players Every aturday, the father plays one set against each of his two sons Over the last two years, Dad has won 70 of 00 sets against the younger son and 50 of 00 sets against the older son Closer examination shows that Dad has won both sets on 30 occasions re the outcomes of the tennis sets independent events? Let event represent Dad beating the younger son and event B represent Dad beating the older son Then, ( B) ( B) 03 ( B) = = = ( ) In general, Dad has a 70% chance of beating the younger son, but he s only got a 60% chance of beating the younger son if he beats the older son that same day Why? Who knows? We don t have to establish a causal (ie, cause-and-effect) relationship between events and B to determine, mathematically, that they are dependent on one another We simply state that the odds of beating the younger son depend on whether or not he beats the older son 8

5 BIT OF HORTHND In many statistics textbooks, you ll see ( B ) written simply as (B) which has its origins in the multiplication rule stated above Likewise, you ll see ( B ) written as (+B) in reference to the addition rule These are just shorthand symbols for union or intersection not mathematical equations ( B) ( and B) ( B ) ( B) ( or B) ( B ) + Using this new shorthand, we can rewrite the basic axioms of probability as: Negation: ( ) = ( ) Union (mutually exclusive events): ( + B) = ( ) + ( B) Union (general): ( + B) = ( ) + ( B) ( B) Conditional robability: ( B) = ( B) ( B) Intersection (independent events): ( B) = ( ) ( B) Intersection (general): ( B) = ( ) ( B) OLVING ROBBILITY ROBLEM We ll see that the words and and or are pretty much magic words when solving probability and statistics problems Whenever you see the word and you want to immediately ask yourself if the outcomes are independent If they are, you can simply multiply their probabilities Likewise, whenever you see the word or you want to immediately ask yourself if the events are mutually exclusive If they are, you can simply add their probabilities 9

6 OME ILLUTRTIVE EXMLE uppose that on the southbound approach to an intersection, 30% of the cars, on the average, turn right lso suppose that 5% of all cars on the road are red What is the probability that the next car approaching the intersection is red and turns right? olution Define a trial as a southbound car approaching the intersection (we'll later call this an "arrival") Define event E as those trials having an outcome of red car and let ( E ) = 0 05 Define event E as those trials having an outcome of turns right and let ( E ) = 0 30 We want to know ( red and right ) which we can write as ( ) = ( ) = ( ) ( ) E E E E E E E Note that we used the word and just now We should immediately ask ourselves whether or not the two events are independent If they are, our calculations get much easier because ( E E ) ( ) = E o, does the probability of a car turning right have anything to do with its color? Lacking any data to the contrary, it s probably safe to assume the two outcomes are completely independent of one another, so we can solve the problem as ( E E ) = ( E ) ( E ) = = If you toss a die, what is the probability of getting either a 4 or a 6? olution Define a trial as a roll of the die Define event E as those trials having an outcome of 4 and let ( E ) = 6 Define event E as those trials having an outcome of 6 and let ( E ) = 6 We want to know ( 4 or 6 ) which we can write as ( ) = ( + ) = ( ) + ( ) ( ) E E E E E E E E We ve just used our other magic word, or, so we should immediately ask ourselves whether or not the two outcomes are mutually exclusive Can a die come up 4 and 6 on the same toss? No ( E + E ) = ( E ) + ( E ) = + = = 3 0

7 3 uppose that on the southbound approach to an intersection, 30% of the cars, on the average, turn right lso suppose that 5% of all cars are red, and that 80% have only one occupant What is the probability that the next car approaching the intersection is a red one with more than one occupant that does not turn right? olution Define a trial as a southbound car approaching the intersection as before Define E = red car and let ( E ) = 0 05 Define E = not turns right and let ( E ) = 0 30 = 0 7 Define E 3 = not single occupant and let ( E ) = 0 80 = 0 3 We want ( red and not turns right and not single occupant ), which can be written as ( E E E )? = 3 We haven t had a problem with three events yet What do we do? Well, we know we can calculate (E E ) as the probability that the next car is red but doesn t turn right: ( E E ) = ( E ) ( ) E red car not turning right is itself an event We can call it E 4 : E = 4 EE With this substitution, our original problem statement can be written as ( E E )? = 4 3 Does the fact that the car is red and doesn t turn right tell you how many people are in the car? No But we can also write ( E E ) = ( E ) ( ) E3 ( E ) = ( E E ) = ( E ) ( ) 4 E which means our original problem can be solved as ( E E E ) = ( E ) ( E ) ( E ) = = 3 3 s you surely suspect by now, regardless of the number of events, if they are independent, we can find the probability of all of them occurring by simply multiplying their probabilities imilarly, if all the events are mutually exclusive, we can find their union by summing their probabilities

8 4 Orders for a certain type of lighting fixture have been summarized according to the optional features that are requested for it: no optional features = 03 one optional feature = 05 more than one option = 0 a What is the probability that an order includes at least one optional feature? b What is the probability that an order includes no more than one optional feature?

9 5 Fifteen of every 400 people is colorblind Fourteen of those are men and one is a woman ssume men make up half the population a What is the probability of being colorblind? b What is the probability of being a colorblind male? c What is the probability of being colorblind IF you are a male? 3