The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Lecture 10 Randomness and Probability Model 5 23 11 Lecture 10 1 Randomness Probability We call a phenomenon or an experiment random if individual outcomes are uncertain but a regular distribution of outcomes emerges with a large number of repetitions Example Toss a coin gender of new born baby The probability of any outcome in a random experiment is approximately the proportion of times the outcome would occur in a very long series of independent repetitions using a long term relative frequency to realize it In the early days probability was associated with games of chance gambling 5 23 11 Lecture 10 2 Probability as long term relative frequency two of many possible sample paths 5 23 11 Lecture 10 3 Probability Model Probability models attempt to model random behavior Consist of two parts A list of possible outcomes sample space S An assignment of probabilities P to each outcome The probability of an event A denoted by P A can be considered as the long run relative frequency of the event A 5 23 11 Lecture 10 4 Sample Space and Events Sample space S the set of all possible outcomes in a random experiment Examples Toss a coin Record the side facing up S Heads Tails H T Toss a coin twice Record the side facing up each time S Toss a coin twice Record the number of heads in the two tosses S Event An outcome or a set of outcomes in a random experiment i e a subset of the sample space 5 23 11 Lecture 10 5 Sample Spaces Events Sample Space a sample space of a random experiment is the set of all possible outcomes Event Simple events The individual outcomes are called simple events 5 23 11 Our objective is to determine P A the probability that an event A will occur Lecture 10 An event is a collection of one or more simple events 6 Toss a coin 3 times Sample space S HHH HHT HTH HTT THH THT TTH TTT There are 8 simple events among which are E1 HHH and E8 TTT Some compound events include A at least two heads HHH HHT HTH THH B exactly two tails 5 23 11 Lecture 10 7 Boy or girl An experiment in a hospital consists of recording the gender of each newborn infant until the birth of a male is observed The sample space of this experiment is S M FM FFM FFFM The sample space contains an infinite number of outcomes 5 23 11 Lecture 10 8 Basic Concepts The complement of an event A the set of all outcomes in S that are not in A not A The union of two events A and B the event consisting of all outcomes that are either in A or in B or in both A B The intersection of two events A and B the event consisting of all outcomes that are in both events A B When two evens A and B have no outcomes in common they are said to be disjoint or mutually exclusive events 5 23 11 Lecture 10 9 Venn Diagram 5 23 11 Lecture 10 10 Probability Rules For any event A 0 P A 1 P S 1 If A and B are disjoint events then P A B P A P B addition rule for disjoint events For any event A P not A 1 P A complement rule For any two events A and B P A B P A P B P A B general addition rule If A and B are disjoint then P A B 0 5 23 11 Lecture 10 11 Equally Likely Outcomes If there are k equally likely outcomes then the probability assigned to each outcome is 1 k P A of outcomes in A k Key smart counting no omission no duplication 5 23 11 Lecture 10 12 Birthday problem In a group of 5 people what is the chance that at least two of them share the same birthday Assume 365 days in one year then the chance is 1 p Detailed work will be shown on the board 5 23 11 Lecture 10 13 Roll a fair die once The label facing up when a fair die is rolled is observed Sample Space S 1 2 3 4 5 6 Every outcome is equally likely to occur P 1 P 2 P 6 1 6 2 3 1 4 5 23 11 6 Venn Diagram 5 Lecture 10 14 Roll a fair die once Consider the following events A The label observed is at most 2 B The label observed is an even number C Label 4 turns up Find P A P not A P A and B P A or C P A or B 5 23 11 Lecture 10 15 Cards A card is drawn from an ordinary deck of 52 playing cards What is the probability that the card is a club a king a club and a king a club or a king neither a club nor a king 5 23 11 Lecture 10 16 Glasses In a group of 88 people in STOR 155 11 out of 50 women and 8 out of 38 men wear glasses What is the probability that a person chosen at random from the group is a woman or someone who wears glasses 5 23 11 Lecture 10 17 Venn diagram with 3 events A Google stock moves up today B Walmart stock moves up today C Exxon stock moves up today P A 0 1 P B 0 2 P C 0 5 P A B 0 05 P A C 0 04 P B C 0 02 P A B C 0 01 Find i P at least one of the 3 stocks go up ii P both Google and Exxon go down iii P only one of the 3 sticks goes up 5 23 11 Lecture 10 18 Continued How to complete a Venn diagram Insert a probability in each disjoint part inside out See details on the board 5 23 11 Lecture 10 19 Take Home Message sample space outcome event union or intersection and complement not disjoint Venn diagram Basic rules For any event A P not A 1 P A If A and B are disjoint then P A B 0 For any two events A and B P A B P A P B P A B 5 23 11 Lecture 10 20
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