5.1

Probability Rules!

PROBABILITY

● 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

LONG TERM PROBABILITY

● 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

DEFINITIONS

● 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”

DEFINITIONS CONTINUED

● 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 e2 = “roll a 2”, e4 =

“roll a 4”, and e6 = “roll a 6” … we’ll write that as {2, 4, 6}

DIE ROLLING

Summary of the example The experiment is rolling a die There are 6 possible outcomes, e1 =

“rolling a 1” which we’ll write as just {1}, e2 = “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}

PROBABILITY…BETWEEN 0 AND 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)

SUM OF PROBABILITIES

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?)

SPECIAL EVENTS

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)

THAT’S 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

PROBABILITY

● 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)

experimenttheoftrialsofnumberoffrequency

)(E

EP

PROBABILITY

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%

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

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

EQUALLY LIKELY OUTCOMES

● 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

CLASSICAL PROBABILITY

● 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)

n

mEP )(

CLASSICAL PROBABILITY

● 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

EXAMPLE

● 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?

TREE DIAGRAM

● 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

TREE DIAGRAM

● 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

EXAMPLE

That leads to two possible outcomes

Michael

Dana

Second ticket

Katherine

Michael

Dana

Start

First ticket

KatherineMichael

KatherineDana

Outcomes

EXAMPLE

We can fill out the rest of the tree What’s the ProbabilityThat Katherine Gets a ticket?

KatherineMichael

KatherineDana

MichaelKatherine

MichaelDanaDana

DanaKatherine

DanaMichael

Katherine

Michael

Katherine

Michael

Dana

Start

Katherine

Michael

Dana

SUBJECTIVE PROBABILITY

● 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 …

SUMMARY

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