Probability Experiments Outcomes Events and Samples Spaces

hey everybody today we're covering the basics of probability we're talking about sample spaces outcomes events and so forth probability experiment or random experiment is trial for which the outcome can't be predicted with certainty although if you run the trial over and over and over again certain trends may emerge here's few examples flip coin and record whether the result is head or tail use random dialer to collect contact 10 random voters and ask whom they intend to vote for roll two dice and record the sum roll two dice and count the number of sixes by the way notice in this last in these last two cases we're doing the same action but the data that we're recording is slightly different so we have different probability experiments little bit more vocabulary the result of particular trial of probability experiment is called the outcome the collection of all possible outcomes of probability experiment is called the sample space or outcome space usually denoted with capital and subset of the sample space is called an event think this all makes more sense when we do an example so here we go let's flip two coins and just record the results the sample space consists of four outcomes heads heads heads tails tails heads and tails tails if we let be the event both flips are the same then we have two outcomes in that event heads heads tails tails this is subset of the sample space roughly speaking an event is something that can happen when you run the probability experiment but there may be multiple different ways that it can happen as in this last example the event both flips are the same can happen two different ways if you have an event that can only happen one way in other words the event consists of single outcome we call that simple event the complement of an event usually denoted prime sometimes with bar over it is the set of all outcomes in the sample space that aren't in so if occurs then prime does not occur and if prime occurs then does not for instance suppose we use spinner to randomly select an integer from 1 to 9 and let be the event the result is prime number here's the sample space it's just the integers from one to nine and here's the event written as set it's all the prime numbers that are less than ten two three five and seven so the complement of prime is going to be the event that does not occur so written as set it's going to be all the numbers less than 10 that are not prime 1 4 6 8 and 9. two events are disjoint if they don't have any outcomes in common to say it differently if they can't both occur for example flip four coins and record the results and let be the event the first two flips are heads and there are at least three tails so the probability experiment consi has sample space that consists of 16 possible outcomes things like heads heads heads heads and heads heads heads tails and can be written as follows the event the first two flips are heads has four outcomes in that event things like heads heads heads heads and heads heads heads tails and at least that there are at least three tails can be written like this here we have five outcomes four where we have three tails and fifth where we have four tails notice that there's no overlap between these two sets there's no outcome that lies in both of the sets these are disjoint to say it in words it's impossible for both of these events to happen at the same time in one trial of this probability experiment now there are actually several different ways of describing the probability of an event all of which we're responsible for first is empirical probability or statistical probability and that's probability just based on observation basically we run probability experiment over and over and over again we count the number of times the event occurs and divide by the total number of times we ran the tr the ran that we ran the probability experiment this is just going to correspond to the proportion of times the event has occurred in the past for instance if we flip coin 100 times and it comes up heads 53 times the empirical probability of the coin coming up heads is 53 and 100 or 53 percent if poll of seven in po now suppose we do poll of 70 of 750 randomly selected pet owners if 412 of them prefer dogs to cats then the empirical probability that pet owner prefers dogs to cats is 412 divided by 750 about 54.9 percent different way of computing probability is classical probability also known as theoretical probability and this only applies when all of the outcomes in sample space are equally likely and in that case we count the number of outcomes in the event count the number of outcomes in the sample space and divide to get the probability to say this little more technically we're doing the cardinality of the set divided by the cardinality of the set so cardinality just means the number of elements in that set here's the notation sort of absolute value of divided by absolute value of and that just means number of outcomes in each couple of examples roll fair die you've got six possible outcomes all equally likely if we're looking at the simple event of getting five on that single roll there's one outcome in that event so the classical probability is going to be one and six if we flip fair coin three times we have eight possible outcomes listed here all of them equally likely let's let be the event that we get exactly two heads there's three outcomes in that event heads heads tails heads tails heads and tails heads heads so the classical probability of the event is going to be three divided by eight there's three equally likely ways of getting exactly two heads out of eight equally likely outcomes possible in total let's run through an example here we have frequency distribution for an introductory statistics class at large university 67 freshmen 72 sophomores and so on what's the probability that we randomly select person from this class and get this is classical probability question there are 222 outcomes total 222 students in the class and 72 outcomes that are in the event that we're interested in that we get sophomore when we select the person at random so the probability of randomly selecting sophomore is 72 divided by 222 about 32.4 percent let's stick with the same frequency distribution and ask slightly different question what's the probability that the next person who registers for the for the course will be either junior or senior this time we're asking for empirical probability we have data about students that have already registered but we're interested in the next person who will register we can't know with certainty what that probability will be however the calculation is going to be basically the same we're going to compute the number of students that have already registered that are juniors or seniors so 29 plus 54 and divide by the total number of students that have already registered to get 37.7 percent although the computation is the same the philosophy behind it is little different this time our probability calculation is based on observation so it's empirical probability not classical regardless of whether we're talking about empirical or classical probability few facts are always going to be consistent first of all for any event the probability is going to be between 0 and 1. 0 is going to be an impossible event and 1 is going to be certain event if is the sample space then the probability of occurring is 1. something is going to happen when you run probability experiment if and are disjoint events with no outcomes in common then the probability of at least one of them occurring is the sum of the probabilities that each one will occur and the probability of them both occurring of course is going to be 0 by the definition of disjoint additionally if we have complementary events then the sum of the probabilities is going to be one one of the two things is going to occur to say that slightly differently the probability that event that event does not occur is one minus the probability that event does occur in our everyday lives we frequently use the language of probability very loosely just estimating the chances that something will happen based on intuition and general life experience in statistics we call that subjective probability for instance might say i'm 99 that i'll win i'm 99 certain that i'll win this game or there's 50 50 chance that my phone will die today while statements like this are descriptive and useful in our regular lives they don't have mathematical meaning and we don't want to base statistical calculations on them this sort of subjective probability is not really the subject of statistics the study of statistics