Work Unit 1 Notes - Introduction To Probability

Work unit 1 Introducing probability ~15 February 2023 Random experiment A procedure whose outcome (result) in a particular performance (triel) connot be predetermines Fair die Equal likelihood of landing on any side they didn't have this property we couldn't use them as random number generators Pr (an an outcome number of (equally likely) ways of getting that outcome number of possible (equally likely) outcomes Pr() - 1/6 Pr (even) 3/6 Odds Reflect the number of ways that give you the event of interest relative to the number of ways that don't give you the event of interest eg Pr () 1/6 odds of (6) 1/5 Zandom variation implies we won't hit a 6 ance every 6 throwS Statistical distributions in this course we will be considering a range of different statistical distributions, which can be applied to different practical situations. Excel Formula RANDBETWEEN interval) (start of interval, end of interval le (1,6) The fair game no one is expected so wen and no one is expected to lose in the long run You bet R1 on a number and you bet 6 times R16 For this game to be foir, we need to payout Ro win for every Practically speaking, Therefore ont 1 bet not a good idea, you would make no money for local communny bot costs 21, but win RS, So instead of Re Expectation 1/6 throws WIN! That retained soc is referred to as house advantage 100 % Win percentage tabal payout for a winning number amount to be bet across all numberss X 5,50 x 100 91,67% house advantage CH 2: Set theory Why do we have to do set theory? One of Murphy's Laws state that before we can do anything, you have to do something else Before we can do "Statistics' we have to do probability theory, and for that we need some "set theory" Definition of sets We define a set A to be a collection of distinguishable objects or entities The set A is determined when we can either (a) list the objects that belong to A or (b) give a rule by which object belongs to A we can decide whether or not a given A= [e, fig the letters e, f, g belong to the set A B = { x 1 1 5 x ≤10} the set B consists of all real numbers X such that I is larger than or equal (keep Soc. For community) to 1 but is less than or equal to 10. e EA e is an element of A ef B e is not an element of B C. 1,3,5,a and D. a,1,5,33 CD the order in which we list the elements of a set is irrelevant E-{a,b,c, a EF and F. a, b.c the set E contains only the distinguisable elements a, b, c. Subsets Suppose we have two gets G and H also belongs to H. Then we say that and that every element of G Gis a subset of H and we write GCH. We can also write H G and say "H Contains G If every element in G does not also belong and say G & not a subseb of H to t, we write G & H Let G-1,3,53, H1,3,5,9} and J. {1,2,3,4,5} GCH, HAJ JOG Note The notation C, for sets is analogous (comparable) to the notat on for ordinary numbers. The "round end of the subset notation tells which of the sets is "smaller (same way the pointed end shows which number is smaller). IF HCG and GCH then obviously, H.G. Intersections and MEL. Suppose L= (a,b,c} But if we consider the set N° and M. {b,c,d) Then L&M {b,c} then we see that NCL and NCM, and that no other sot of which N is a subset has this property. This leads us to the idea of intersection. The intersection of any two sets is the set that contains precisely those elements which belong to both sets. For the sets L, M, N above we write N-LAM and read this 'N equals L intersection M. The intersection of two sets M and N can be thought of as the set containing those elements which belong to bath M and to N. The empty set, mutually exclusive sets. What happens if L. a,b,c} and 2- Ed,e,f}? If If we want LAR to be a set, then we must introduce a new concept, the empty set, the set that has no members. This is a sensible concept consider the set of English speaking fish, or consider the set of real numbers whose square is negative We reserve the symbol to denote the empty set. We use this we write and read this as LOR = Ø symbol for no other purpose the intersection of sets Land P is the empty set Pairs of sets whose intersection is the empty set are said to be mutually exclusive sets (or disjoint sets) Thus Land R are mutually exclusive. The universal set, the sample space. Anather reserved symbol is the letter S. It is used for the set containing all objects under consideration. Thus if, in a particular problem, the only objects of interest are the colours of the traffic lights, then S. {red &red red, amber, green The S set is known to mathematicians as the universal sel. In statistical Jargon the set S is called the sample space Unions The concept union contrasts with the concept intersection. The union of two sees A and B is the Set that contains the elements that A or B Here we use the word or in belong to an inclusive sense we do not exclude from the union those elements that belong to both A and B. If A= 1,2,33 is the set and B. 2,3,4,5] C1,2,3,4,5} 1, 2, 3, 4, 53 we write then the union of A and B C- AUB we say C equals A union B