advertisement

CP Probability & Statistics SP 2015 Our eventual goal in this course is to go from the random sample to the population. The theory that allows for this transition is the theory of probability. Random Phenomenon We know what outcomes could happen, but not which outcomes will happen. The list of possible outcomes of a random experiment must be exhaustive and mutually exclusive. Examples: Experiment Flip a coin Roll a standard die Outcomes Heads, Tails 1, 2, 3, 4, 5, 6 Probability of an event A, P(A), is a number between 0 and 1 that identifies the likelihood that event A happens. Example: Rolling a standard die P(1) = 1/6 P(2) = 1/6 P(3) = 1/6 P(4) = 1/6 P(5) = 1/6 P(6) = 1/6 Trial Single attempt (or realization) of a random phenomenon Outcome The observed result of a trial Independence (informal definition) 2 events are independent if the outcome of 1 does not influence the outcome of the other. Event Collection of outcomes We typically label events so we can attach probabilities to them Notation: bold capital letter: A, B, C, … Sample Space The collection of all possible outcomes Outcomes O1, O2, . . . Sample space S Thus, in set-theory notation, S = {O1, O2,. ..} Sample Space: S = {1, 2, 3, 4, 5, 6} Trial: Roll the die Outcome: “3” Events: Outcome is an even number (one of 2, 4, 6) Outcome is a low number (one of 1, 2, 3) P(1) = 1/6 P(4) = 1/6 P(2) = 1/6 P(5) = 1/6 P(3) = 1/6 P(6) = 1/6 Note: The sum of probabilities for a Sample Space is always 1 Law of Large Numbers As the number of trials increases, the longrun relative frequency of repeated independent events gets closer to the true relative frequency (the observed probability gets closer to the calculated/theoretical probability) If event A has m outcome and event B has n outcomes then the number of possible outcomes for A or B is m + n A and B is mn n! = n(n-1)(n-2)…(1) Example: 5! = 5 * 4 * 3 * 2 * 1 Number of possible arrangements when choosing r items from a set of n items and ORDER MATTERS n! P n r (n r )! Number of possible arrangements when choosing r items from a set of n items and ORDER DOES NOT MATTERS n! C n r r !(n r )! When the outcomes in a sample space are equally likely to occur then: number of times A occurs in sample space P( A) number of items in sample space The number of distinct permutations of n objects where n1 of the objects are identical, n2 of the objects are identical, . . . , nr of the objects are identical is found by the formula: n! P n r n1 !n2 !.....nr ! How many different arrangements can be made using all of the letters in MISSISSIPPI M 1 I 4 S 4 P 2 11! 11 P11 4!4!2! 11 10 9 8 7 6 5 4! 11 10 9 8 7 6 5 11 10 3 1 7 3 5 34650 2!4!4! 214 3 21 111111