Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 1 of 33 Chapter 5 Section 1 Probability Rules

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 2 of 33 Chapter 5 – Section 1 ●Learning objectives  Understand the rules of probabilities  Compute and interpret probabilities using the empirical method  Compute and interpret probabilities using the classical method  Use simulation to obtain data based on probabilities  Understand subjective probabilities

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 3 of 33 Chapter 5 – Section 1 ●Learning objectives  Understand the rules of probabilities  Compute and interpret probabilities using the empirical method  Compute and interpret probabilities using the classical method  Use simulation to obtain data based on probabilities  Understand subjective probabilities

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 4 of 33 Chapter 5 – Section 1 ●Probability relates short-term results to long-term results ●An example ●Probability relates short-term results to long-term results ●An example  A short term result – what is the chance of getting a proportion of 2/3 heads when flipping a coin 3 times ●Probability relates short-term results to long-term results ●An example  A short term result – what is the chance of getting a proportion of 2/3 heads when flipping a coin 3 times  A long term result – what is the long-term proportion of heads after a great many flips ●Probability relates short-term results to long-term results ●An example  A short term result – what is the chance of getting a proportion of 2/3 heads when flipping a coin 3 times  A long term result – what is the long-term proportion of heads after a great many flips  A “fair” coin would yield heads 1/2 of the time – we would like to use this theory in modeling

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 5 of 33 Chapter 5 – Section 1 ●Relation between long-term and theory  The long term proportion of heads after a great many flips is 1/2  This is called the Law of Large Numbers ●Relation between long-term and theory  The long term proportion of heads after a great many flips is 1/2  This is called the Law of Large Numbers ●Relation between short-term and theory  We can compute probabilities such as the chance of getting a proportion of 2/3 heads when flipping a coin 3 times by using the theory  This is the probability that we will study

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 6 of 33 Chapter 5 – Section 1 ●Some definitions  An experiment is a repeatable process where the results are uncertain ●Some definitions  An experiment is a repeatable process where the results are uncertain  An outcome is one specific possible result ●Some definitions  An experiment is a repeatable process where the results are uncertain  An outcome is one specific possible result  The set of all possible outcomes is the sample space ●Some definitions  An experiment is a repeatable process where the results are uncertain  An outcome is one specific possible result  The set of all possible outcomes is the sample space ●Example  Experiment … roll a fair 6 sided die  One of the outcomes … roll a “4”  The sample space … roll a “1” or “2” or “3” or “4” or “5” or “6”

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 7 of 33 Chapter 5 – Section 1 ●More definitions  An event is a collection of possible outcomes … we will use capital letters such as E for events  Outcomes are also sometimes called simple events … we will use lower case letters such as e for outcomes / simple events ●More definitions  An event is a collection of possible outcomes … we will use capital letters such as E for events  Outcomes are also sometimes called simple events … we will use lower case letters such as e for outcomes / simple events ●Example (continued)‏  One of the events … E = {roll an even number}  E consists of the outcomes e 2 = “roll a 2”, e 4 = “roll a 4”, and e 6 = “roll a 6” … we’ll write that as {2, 4, 6}

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 8 of 33 Chapter 5 – Section 1 ●Summary of the example  The experiment is rolling a die  There are 6 possible outcomes, e 1 = “rolling a 1” which we’ll write as just {1}, e 2 = “rolling a 2” or {2}, …  The sample space is the collection of those 6 outcomes {1, 2, 3, 4, 5, 6}  One event is E = “rolling an even number” is {2, 4, 6}

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 9 of 33 Chapter 5 – Section 1 ●If E is an event, then we write P(E) as the probability of the event E happening ●These probabilities must obey certain mathematical rules ●We will be studying varying classes of probabilities … these rules are true for all of them

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 10 of 33 Chapter 5 – Section 1 ●Rule – the probability of any event must be greater than or equal to 0 and less than or equal to 1  It does not make sense to say that there is a –30% chance of rain  It does not make sense to say that there is a 140% chance of rain ●Note – probabilities can be written as decimals (0, 0.3, 1.0), or as percents (0%, 30%, 100%), or as fractions (3/10)‏

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 11 of 33 Chapter 5 – Section 1 ●Rule – the sum of the probabilities of all the outcomes must equal 1  If we examine all possible cases, one of them must happen  It does not make sense to say that there are two possibilities, one occurring with probability 20% and the other with probability 50% (where did the other 30% go?)‏

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 12 of 33 Chapter 5 – Section 1 ●Probability models must satisfy both of these rules ●There are some special types of events  If an event is impossible, then its probability must be equal to 0 (i.e. it can never happen)‏  If an event is a certainty, then its probability must be equal to 1 (i.e. it always happens)‏

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 13 of 33 Chapter 5 – Section 1 ●A more sophisticated concept  An unusual event is one that has a low probability of occurring  This is not precise … how low is “low? ●A more sophisticated concept  An unusual event is one that has a low probability of occurring  This is not precise … how low is “low? ●Typically, probabilities of 5% or less are considered low … events with probabilities of 5% or lower are considered unusual ●A more sophisticated concept  An unusual event is one that has a low probability of occurring  This is not precise … how low is “low? ●Typically, probabilities of 5% or less are considered low … events with probabilities of 5% or lower are considered unusual ●However, this cutoff point can vary by the context of the problem

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 14 of 33 Chapter 5 – Section 1 ●Learning objectives  Understand the rules of probabilities  Compute and interpret probabilities using the empirical method  Compute and interpret probabilities using the classical method  Use simulation to obtain data based on probabilities  Understand subjective probabilities

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 15 of 33 Chapter 5 – Section 1 ●If we do not know the probability of a certain event E, we can conduct a series of experiments to approximate it by ●This becomes a good approximation for P(E) if we have a large number of trials (the law of large numbers)‏

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 16 of 33 Chapter 5 – Section 1 ●Example ●We wish to determine what proportion of students at a certain school have type A blood  We perform an experiment (a simple random sample!) with 100 students  If 29 of those students have type A blood, then we would estimate that the proportion of students at this school with type A blood is 29%

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 17 of 33 Chapter 5 – Section 1 ●Example (continued)‏ ●We wish to determine what proportion of students at a certain school have type AB blood  We perform an experiment (a simple random sample!) with 100 students  If 3 of those students have type AB blood, then we would estimate that the proportion of students at this school with type AB blood is 3%  This would be an unusual event

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 18 of 33 Chapter 5 – Section 1 ●Learning objectives  Understand the rules of probabilities  Compute and interpret probabilities using the empirical method  Compute and interpret probabilities using the classical method  Use simulation to obtain data based on probabilities  Understand subjective probabilities

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 19 of 33 Chapter 5 – Section 1 ●The classical method applies to situations where all possible outcomes have the same probability ●This is also called equally likely outcomes ●The classical method applies to situations where all possible outcomes have the same probability ●This is also called equally likely outcomes ●Examples  Flipping a fair coin … two outcomes (heads and tails) … both equally likely  Rolling a fair die … six outcomes (1, 2, 3, 4, 5, and 6) … all equally likely  Choosing one student out of 250 in a simple random sample … 250 outcomes … all equally likely

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 20 of 33 Chapter 5 – Section 1 ●Because all the outcomes are equally likely, then each outcome occurs with probability 1/n where n is the number of outcomes ●Examples  Flipping a fair coin … two outcomes (heads and tails) … each occurs with probability 1/2 ●Because all the outcomes are equally likely, then each outcome occurs with probability 1/n where n is the number of outcomes ●Examples  Flipping a fair coin … two outcomes (heads and tails) … each occurs with probability 1/2  Rolling a fair die … six outcomes (1, 2, 3, 4, 5, and 6) … each occurs with probability 1/6 ●Because all the outcomes are equally likely, then each outcome occurs with probability 1/n where n is the number of outcomes ●Examples  Flipping a fair coin … two outcomes (heads and tails) … each occurs with probability 1/2  Rolling a fair die … six outcomes (1, 2, 3, 4, 5, and 6) … each occurs with probability 1/6  Choosing one student out of 250 in a simple random sample … 250 outcomes … each occurs with probability 1/250

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 21 of 33 Chapter 5 – Section 1 ●The general formula is ●If we have an experiment where  There are n equally likely outcomes (i.e. N(S) = n)‏ ●The general formula is ●If we have an experiment where  There are n equally likely outcomes (i.e. N(S) = n)‏  The event E consists of m of them (i.e. N(E) = m)‏ ●The general formula is ●If we have an experiment where  There are n equally likely outcomes (i.e. N(S) = n)‏  The event E consists of m of them (i.e. N(E) = m)‏ then

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 22 of 33 Chapter 5 – Section 1 ●Because we need to compute the “m” or the N(E), classical methods are essentially methods of counting ●These methods can be very complex! ●An easy example first ●Because we need to compute the “m” or the N(E), classical methods are essentially methods of counting ●These methods can be very complex! ●An easy example first ●For a die, the probability of rolling an even number ●Because we need to compute the “m” or the N(E), classical methods are essentially methods of counting ●These methods can be very complex! ●An easy example first ●For a die, the probability of rolling an even number  N(S) = 6 (6 total outcomes in the sample space)‏ ●Because we need to compute the “m” or the N(E), classical methods are essentially methods of counting ●These methods can be very complex! ●An easy example first ●For a die, the probability of rolling an even number  N(S) = 6 (6 total outcomes in the sample space)‏  N(E) = 3 (3 outcomes for the event)‏ ●Because we need to compute the “m” or the N(E), classical methods are essentially methods of counting ●These methods can be very complex! ●An easy example first ●For a die, the probability of rolling an even number  N(S) = 6 (6 total outcomes in the sample space)‏  N(E) = 3 (3 outcomes for the event)‏  P(E) = 3/6 or 1/2

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 23 of 33 Chapter 5 – Section 1 ●A more complex example ●Three students (Katherine, Michael, and Dana) want to go to a concert but there are only two tickets available ●A more complex example ●Three students (Katherine, Michael, and Dana) want to go to a concert but there are only two tickets available ●Two of the three students are selected at random ●A more complex example ●Three students (Katherine, Michael, and Dana) want to go to a concert but there are only two tickets available ●Two of the three students are selected at random  What is the sample space of who goes?  What is the probability that Katherine goes?

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 24 of 33 Chapter 5 – Section 1 ●Example continued ●We can draw a tree diagram to solve this problem Katherine Michael Dana Start ●Example continued ●We can draw a tree diagram to solve this problem ●Who gets the first ticket? Any one of the three … First ticket

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 25 of 33 Chapter 5 – Section 1 ●Who gets the second ticket? Michael Dana Katherine Michael Dana Start First ticket ●Who gets the second ticket?  If Katherine got the first, then either Michael or Dana could get the second Second ticket

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 26 of 33 Chapter 5 – Section 1 ●That leads to two possible outcomes Michael Dana Second ticket Katherine Michael Dana Start First ticket Katherine Michael Katherine Dana Outcomes

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 27 of 33 Chapter 5 – Section 1 ●We can fill out the rest of the tree KatherineMichaelDanaStart Katherine Michael Katherine Dana Katherine Michael Katherine Michael Dana MichaelDana Katherine Dana Michael KatherineMichaelDana

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 28 of 33 Chapter 5 – Section 1 ●Katherine goes in 4 out of the 6 outcomes … a 4/6 or 2/3 probability Katherine Michael Dana Start Katherine Michael Katherine Dana Katherine Michael Katherine Michael Dana Michael Dana Katherine Dana Michael Katherine Michael Dana

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 29 of 33 Chapter 5 – Section 1 ●Learning objectives  Understand the rules of probabilities  Compute and interpret probabilities using the empirical method  Compute and interpret probabilities using the classical method  Use simulation to obtain data based on probabilities  Understand subjective probabilities

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 30 of 33 Chapter 5 – Section 1 ●Sometimes probabilities are difficult to calculate, but the experiment can be simulated on a computer ●If we simulate the experiment multiple times, then this is similar to the situation for the empirical method ●We can use

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 31 of 33 Chapter 5 – Section 1 ●Learning objectives  Understand the rules of probabilities  Compute and interpret probabilities using the empirical method  Compute and interpret probabilities using the classical method  Use simulation to obtain data based on probabilities  Understand subjective probabilities

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 32 of 33 Chapter 5 – Section 1 ●A subjective probability is a person’s estimate of the chance of an event occurring ●This is based on personal judgment ●A subjective probability is a person’s estimate of the chance of an event occurring ●This is based on personal judgment ●Subjective probabilities should be between 0 and 1, but may not obey all the laws of probability ●For example, 90% of the people consider themselves better than average drivers …

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 33 of 33 Summary: Chapter 5 – Section 1 ●Probabilities describe the chances of events occurring … events consisting of outcomes in a sample space ●Probabilities must obey certain rules such as always being greater than or equal to 0 ●There are various ways to compute probabilities, including empirically, using classical methods, and by simulations

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 34 of 33 Examples ●Identify as Classical, Empirical, or Subjective Probability: In his fall 1998 article in Chance Magazine, (“A Statistician Reads the Sports Pages,” pp ,) Hal Stern investigated the odds that a particular horse will win a race. He reports that these odds are based on the amount of money bet on each horse. The odds can be used to calculate probabilities. When a probability is given that a particular horse will win a race, is this empirical, classical, or subjective probability?

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 35 of 33 Examples ●Identify as Classical, Empirical, or Subjective Probability: Pass the Pigs TM is a Milton-Bradley game in which pigs are used as dice. Points are earned based on the way the pigs land. There are six possible outcomes when one pig is tossed. A class of 52 students rolled pigs 3,939 times. The number of times each outcome occurred is recorded in the table at right. (Source: Are these probabilities empirical, classical, or subjective?

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 36 of 33 Examples ●Identify as Classical, Empirical, or Subjective Probability: In a draft lottery, balls representing each birthday are placed in a bin and mixed. Individuals whose birth date is drawn are selected for military service. Ignore leap year. The probability that a particular day, i.e. July 1, will be selected on the first draw is 1/365. Is this an example of an empirical, classical, or subjective probability?

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 37 of 33 ●Suppose two students are selected at random. ●What is the probability that the first student was born in April? ●What is the probability that both students were born in July? ●Is it likely that these two students share the same birthday?

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 5 Section 1 – Slide 38 of 33 ●30/365 ≈ ●961/ ≈ ●