o Varies from Statistics Statistics uses sample information to make inference to what is occurring in the BIOM301 Chapter 4 Probability Probability Uses population info to make predictions about samples population Definitions o Experiment process that yields 1 random result or observation o Outcomes all the possible results o Event ONE outcome of interest o Probability the measure of how likely an event is 3 Ways to Find the Probability of an Event Occurring Empirically Used when it is not obvious what the result should be o Often look at Long Term Behavior o Relative Frequency vs Cumulative Relative Frequency o Law of Large Numbers With repetition empirical results will approach the expected theoretical probability which we haven t defined yet Subjectively Based on Personal Judgment o Ex Meteorologists using their expert knowledge to estimate the probability of rain o No special symbols just stated as subjective probability Theoretically Probability that a random event will occur through reasoning or calculation o Ex The probability of rolling a die and getting the number 5 should be 1 6 since there are six sides to a die and each are equally likely o Theoretical probability P A where A is some random outcome of an experiment o Many Ways to View Probability for Random Events Tree Diagram Shows possible outcomes but not their likelihood Sample Space Does show the probability of each outcome because items in sample space must be equally likely and all outcomes must be shown Venn Diagram List the number of observations that fall into some category so you can calculate probability Contingency Table Can calculate probability from information shown in the table Compound Events Events made up of 1 simple event Ex What are the possible outcomes and their probabilities of the sex and birth order when you have 2 children Next Step Calculating Probability What if P Male P Female o P Male 0 52 o P Female 0 48 o How would you calculate the probability of having 2 children and they are both female Need to Know o Whether we are asking if Event A or Event B occurs If Event A and Event B occurs 2 rules to deal with how these probabilities are calculated o Multiplication Rule AND The probability that 1 simple events occur is equal the probability of each multiplied by the others IF THE EVENTS ARE INDEPENDENT Ex P having 2 children and they are both female o P 1st child is female P 2nd child is also female o 0 48 0 48 Independence When the outcome of the first event has no influence on the outcome of the second event P A and B P A P B Not independent P A and B P A P B I A where P B A P A and B P A This is just a mathematical substitution You must be GIVEN the probability that BOTH occur you don t calculate it yourself by multiplying Are 2 events independent If event A is independent of event B then the P A P A B o Ex If the probability of A is 0 20 and the probability of B is 0 40 and the probability of A and B is 0 08 then is Event A independent of Event B o P A B 0 08 0 4 0 20 which is P A so YES Events A and B are independent In other words If 2 events are independent then the likelihood of event A is the same for the whole population as well as for a subject o Addition Rule OR What about the P A or B occurring When thinking about whether one outcome OR another occurred we are thinking about 2 events that are Mutually Exclusive meaning only 1 event can occur Ex If you have a baby you can only have a Boy OR a Girl To calculate the probability of a mutually exclusive event you add P A P B to find out the P A or B Might seem counterintuitive but since more than 1 outcome can meet the requirements the probability should go up addition o Ex What is the probability that a person in this class is a senior OR a junior General Addition Rule IF the P A or B P A P B then YES events A and B are mutually exclusive P A or B P A P B P A and B If Events are mutually exclusive then P A and B 0 P A or B P A P B P A and B If events are not mutually exclusive the probability of both occurring is included twice By subtracting P A and B you get rid of this double counting Addition Rule can deal with events that are NOT mutually exclusive Are 2 events mutually exclusive The Complement of A probability that Event A does NOT occur Sampling WITHOUT REPLACEMENT You need the change probability of an event as you go Probability statements often made in terms of chances or risks Can also state Probabilities as Odds The odds of your favorite race horse Golden Horseshoe winning the Kentucky Derby are 3 to 1 for P Golden Horseshoe wins Derby 3 1 3 75 Equation Odds for an event are a to b Then probability of event a a b The probability of the complement P A 1 P A There can be more than 2 outcomes The sum of all possible probabilities must sum to 1 0 REMEMBER Don t assume if there are just 2 outcomes that the P A P B 0 5 Multiple ways to think about probabilities o Tree diagrams o Sample space o Venn Diagrams o Contingency Tables