The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Lecture 10 Randomness and Probability Model 10 6 09 Lecture 10 1 The Monty Hall Problem Let s Make A Deal a game show back in the 90 s A player is given the choice of three doors Behind one door is the Grand Prize a car and a cruise behind the other two doors booby prizes stinking pigs The player picks a door and the host peeks behind the doors and opens one of the rest of the doors There is a booby prize behind the open door The host offers the player either to stay with the door that was chosen at the beginning or to switch to the remaining closed door Which is better to switch doors or to stay with the original choice What are the chances of winning in either case 10 6 09 Lecture 10 2 3 Prisoners Dilemma Three prisoners A B and C are on death row The governor decides to pardon one of the three and chooses at random the prisoner to pardon He informs the warden of his choice but requests that the name be kept secret for a few days The next day A tries to get the warden to tell him who had been pardoned The warden refuses A then asks which of B or C will be executed The warden thinks for a while then tells A that B is to be executed Can A increase his chance of survival by swapping his fate with C 10 6 09 Lecture 10 3 Remarks The previous two problems are equivalent Play it online at http www shodor org interactivate activities mont y3 How can we solve similar problems systematically Probability models 10 6 09 Lecture 10 4 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 10 6 09 Lecture 10 5 Probability as long term relative frequency 10 6 09 Lecture 10 6 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 10 6 09 Lecture 10 7 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 10 6 09 Lecture 10 8 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 10 6 09 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 9 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 10 6 09 Lecture 10 10 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 10 6 09 Lecture 10 11 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 10 6 09 Lecture 10 12 Venn Diagram 10 6 09 Lecture 10 13 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 10 6 09 Lecture 10 14 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 10 6 09 Lecture 10 15 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 10 6 09 Lecture 10 16 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 10 6 09 6 Venn Diagram 5 Lecture 10 17 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 10 6 09 Lecture 10 18 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 10 6 09 Lecture 10 19 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 10 6 09 Lecture 10 20 Venn diagram with 3 events A Google stock moves …
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