Chapter 4 Probability 1 Probability uses population info to make predictions about samples 2 Definition Example An experiment process that yields 1 result or observation In the problem above the experiment is spinning the spinner outcomes all the possible results The possible outcomes are landing on yellow blue green or red An event is one outcome of interest We might be interested in landing on blue Probability is the measure of how likely an event is The probability of landing on blue is 1 out of 4 1 4th 3 3 ways to to find the probability of an event occurring a Empirically b Theoretically c Subjectively 4 Empirically Prime notation used for Empirical Probabilities P Probability and P Empirical Probability A specific outcomes can use any letter e g B C etc n A number of times the event A has occurred n total number of times the experiment is attempted Law of Large Numbers expected theoretical probability With repetition empirical results will approach the As 5 Theoretically a Sample Space S a list of all possible outcomes that could occur for some experiment b Determined by your experimental question c Outcomes must all be equally likely d Denoted by S e The number of outcomes in S n S f If all outcomes in a sample space are equally likely then you can calculate the probability of an event as the proportion of times your event occurs out of the total g P A of times event A occurs in sample space of outcomes in sample space h P A n A n S 6 Subjectively a These are based on personal judgement b example Meteorologists using their expert knowledge to estimate the probability of rain 7 The probability of an event is between 0 and 1 a The probability of an impossible event 0 b The probability of a certain event 1 8 The sum of all possible outcomes 1 9 Complement 10 Odds 11 P A or B a Equation Odds for an event are a to b b Then probability of event a a b a The Complement of A probability that Event A does NOT occur b The probability of the complement P A 1 P A a 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 b Example of mutually exclusive if you have a baby you can only have a Boy OR a Girl c To calculate the probability of a mutually exclusive event you add the P A P B to find out the P A or B d Might seem counterintuitive but since more than 1 outcome can meet the requirements the probability should go up addition e P A or B P A P B P A and B f 12 P A and B If events are mutually exclusive then P A and B 0 a P A and B P A P B b 2 events are independent c So to calculate the probability of Independent Events you end up multiplying the probabilities d This makes sense since you need not just 1 but 2 events to occur e Since it is less likely that 2 events will occur the probability should go down 13 Conditional Probabilities a P B A the probability that B will also occur given that A has occurred b P B A P A and B c If event A is independent of event B then the P A P A B P A Chapter 5 Discrete Probability Distributions 1 A random variable is simply a variable x whose value depends on the outcome of a chance operation random variable 2 So for a a Outcome based on chance operation b 1 outcome per run of the chance operation c Every outcome is independent of every other outcome 3 Numerical Random Variables can be divided into 2 groups Discrete Random Variables Chapter 5 quantitative discrete Continuous Random Variables Chapter 6 quantitative continuous 4 Discrete Random Variables Probability Function or Probability Distribution to describe Relationship Chapter 6 Continuous Random Variables 1 Normal Distribution a Often but by no means always continuous random variables have a distribution that is symmetric bell shaped b This is the Normal Distribution c The curve is called a Normal Curve 2 Standard Normal Curve a Total area 1 symmetric and infinite 3 Finding areas with Standard Normal Curve If you are told that a variable is a i Normally distributed with a ii 0 and c 1 then it is the Standard Normal Curve 4 Finding areas with ANY normal curve 5 z notation z is the z score point on the z axis when there is of the area 6 Normal Approximation of a binomial distribution probability to the right of z a Normal distribution provides a reasonable approximation to a binomial probability when n p 5 AND n 1 p 5 b Useful when n is large