Probability Rules!

● 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

● 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

● 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”

● 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}

 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}

 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)

 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?)

 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)

● 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

● 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)

 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 (this is empirical probability since we are collecting our own data)  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%

 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

● 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

● 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

● What is “theoretically supposed to happen” ● 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)

● 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

● 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?

● Example continued ● We can draw a tree diagram to solve this problem ● Who gets the first ticket? Any one of the three… 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 Michael Dana Katherine Michael Dana Start First ticket Second ticket

 That leads to two possible outcomes Michael Dana Second ticket Katherine Michael Dana Start First ticket Katherine Michael Katherine Dana Outcomes

 We can fill out the rest of the tree  What’s the Probability That Katherine Gets a ticket? Katherine Michael Katherine Dana Michael Katherine Michael Dana Katherine Dana Michael KatherineMichael KatherineMichaelDanaStart KatherineMichaelDana

● 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 …

 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