Probability An Introduction Probability is as intimately related to counting processes as your skin is to your skeleton Please back up and read review the sets and counting assignments Although your text puts this in the second section counting is really the first step Without counting we cannot cover the concept except in fuzzy generalities By the end of this section you should believe this We need to recognize some differences between the counting vocabulary and the probability vocabulary Let s begin with a definition of probability Probability is the study of the likelihood or chance that an outcome or event can will or has occurred within given restrictions This sentence is replete with words that need explanation An outcome is some basic result that we can record in a process Example We flip a coin The outcomes are to see heads H or tails T We discount the on edge possibility as a nonevent in most considerations Even in a football game the ref has to toss again Example We flip the coin two times The outcomes are HH HT TH TT Example We spin a dial with areas marked a b c Then the outcomes are a b or c Example We spin the dial two times The outcomes are aa ab ac ba bb bc ca cb cc An outcome space1 usually denoted by S is the collection of all possible outcomes from some action or process Example Say we flip a coin The outcomes are a heads or a tails These are the only two possible outcomes We discount edgies and require a re toss just as in most sporting events S heads tails H T Example Say we flip a coin two times The outcomes are of each spin are heads or tails However the outcome space is composed of the results of both spins We list them as ordered pairs S heads heads heads tails tails heads tails tails HH HT TH TT simpler The number of elements in the outcome space is simply an application of counting methods 2 There are two choices at each flip So there are 2 2 2 4 outcomes Example We spin a dial with areas marked a b c S a b c Example We spin a dial with areas marked a b c three times Similarly to the coin flip the outcomes are not a b or c They are the ordered triples listed as first spin second spin third spin The outcome space is listed in the table below aaa aab aac aba abb abc aca acb acc baa bab bac bba bbb bbc bca bcb bcb caa cab cab cba cbb cbc cca ccb ccc 1 Most text call this a Sample Space which makes the S a sensible letter assignment Arizona State University Department of Mathematics and Statistics 1 of 5 Probability An Introduction Listing the 27 outcomes is tedious Fortunately we are more interested than the count of the set than the list itself in many cases Again counting the outcome space is an application of the multiplication principle 3 There are three choices at each spin The outcome space has 3 3 3 3 27 outcomes A word of caution about outcome spaces These are not universal sets strictly speaking Sets are unordered complete listings of elements with each one listed exactly once As we will see later sometimes it is convenient to allow repetition in an outcome space to better understand the probabilities An event is any combination of outcomes using words like and or not excluding etc Example For the previous example let A be the event where the three character combination includes the letter a The list for A remain in the table below Note there are 19 of them aaa aab aac aba abb abc aca acb acc baa bab bac bba bca caa cab cab cba cca So how do we count them without listing them Since there are 27 possibilities to begin with we should remove the bad ones They are the ones where we use only two letters b or c to create the combination There are 2 2 2 23 8 baddies That leaves 33 23 27 8 19 goodies Likelihood of an Event Calculated Likelihood of an event implies that there should be some quantification or numerical value assigned to a probability Words like certain possible or impossible are used to quantify likelihood To make statements in a more succinct way we adopt the notation that p X stands for the probability that situation X occurs 2 This is function notation since for any specific event X there is exactly one probability of its occurrence We can make the following numerical assignments without any other information Impossible event X simply cannot happen so p X 0 Example We want the likelihood of picking an apple from a pine tree The probability of that silly event is zero zip nada It is impossible in any natural way Call it impossible Certain event X must happen so p X 100 Example We want the likelihood of finding an apple tree in an apple orchard the probability of that event is a certainty We assign the value 100 1 unit The most certain you can be is 100 Notice the wording We did not say finding only apple trees in the orchard or even finding apples in an apple orchard These events are not certain There may be other trees that have been allowed to remain in 2 2 of 5 Your textbook uses a capital P I have chosen to use the lower case p because we also use the capital P for permutation notation and too many other things Arizona State University Department of Mathematics and Statistics Probability An Introduction the orchard but why call it an apple orchard if it doesn t have apple trees Also if it is too early in the growing season there may be no apples present yet Philosophy 101 While it may be encouraging to have someone say they ll give 110 it is impossible You cannot give more than you have period You can however admit you ve been holding out in your previous efforts Shame on you Then you might be able to increase your effort by 110 a factor of 1 10 so long as that value remains below the 100 possible Probability is exactly this way The sum of the probability of all events in a sample space must be 100 Possible event X can happen so 0 p X 100 Whatever the outcome space is we should include it in the thoughts about likelihood Notice that since the event is possible p X 0 Example We want the likelihood that in a group of 367 people two of them have the same birthday By birthday we mean month and day but disregard the year Since there are 366 possible birthdays don t forget leap years we cannot state that this is a certainty However with only two days not covered it is almost inconceivable that the group doesn t have two with matching birthdays Next let s use some simple examples to …