ProbabilityProbability

Probability is a numerical Probability is a numerical measurement of likelihood of an measurement of likelihood of an event.event.

The probability of any event is a The probability of any event is a number between zero and one.number between zero and one.

Events with probability close to one Events with probability close to one are more likely to occur.are more likely to occur.

If an event has probability equal to If an event has probability equal to one, the event is certain to occur.one, the event is certain to occur.

Probability NotationProbability Notation

If A represents an event,If A represents an event,

P(A)P(A)

represents the probability of represents the probability of A.A.

Three methods to find Three methods to find probabilities:probabilities:

IntuitionIntuition

Relative frequencyRelative frequency

Equally likely outcomesEqually likely outcomes

Intuition methodIntuition method

based upon our level of based upon our level of confidence in the resultconfidence in the result

Example: I am 95% sure that Example: I am 95% sure that I will attend the party.I will attend the party.

Probability Probability as Relative Frequencyas Relative Frequency

Probability of an event =Probability of an event =

the fraction of the time that the event the fraction of the time that the event occurred in the past =occurred in the past =

f f nn

where f = frequency of an eventwhere f = frequency of an eventn = sample sizen = sample size

Example of Probability as Relative Frequency

If you note that 57 of the last 100 applicants for a job have been female, the

probability that the next applicant is female would be:

10057

Equally likely outcomesEqually likely outcomes

No one result is expected to No one result is expected to occur more frequently than occur more frequently than

any other.any other.

Probability of an event when Probability of an event when outcomes are equally likely =outcomes are equally likely =

outcomesofnumbertotal

eventtofavorableoutcomesofnumber

Example of Equally Likely Example of Equally Likely Outcome MethodOutcome Method

When rolling a die, the probability of When rolling a die, the probability of getting a number less than three =getting a number less than three =

3

1

6

2

Law of Large NumbersLaw of Large Numbers

In the long run,In the long run,as the sample size increases and as the sample size increases and

increases, the relative frequencies of increases, the relative frequencies of outcomes outcomes

get closer and closer to get closer and closer to the theoretical (or actual) probability the theoretical (or actual) probability

value.value.

Example of Law of Large Example of Law of Large NumbersNumbers

Flip a coin 10 times and record the outcomes. Flip a coin 10 times and record the outcomes. (Do it now…ok 1 person do it…no need for (Do it now…ok 1 person do it…no need for everyone to throw money around!)everyone to throw money around!)

Flip a coin 20 times and record the outcomes. Flip a coin 20 times and record the outcomes. (Yes, do this now also!)(Yes, do this now also!)

Flip a coin 30 times and record the outcomes. Flip a coin 30 times and record the outcomes. (Don’t skip this step!)(Don’t skip this step!)

Example of Law of Large Example of Law of Large NumbersNumbers

What happens to the P(heads) vs P(tails) What happens to the P(heads) vs P(tails) as the sample space increases?as the sample space increases?

How does the actual result compare to the How does the actual result compare to the theoretical result? What did happen vs theoretical result? What did happen vs what should have happened?what should have happened?

Statistical ExperimentStatistical Experiment

activity that results in a activity that results in a definite outcomedefinite outcome

Sample SpaceSample Space

set of all possible outcomes of set of all possible outcomes of an experimentan experiment

Sample Space for the Sample Space for the rolling of an ordinary die:rolling of an ordinary die:

1, 2, 3, 4, 5, 61, 2, 3, 4, 5, 6

For the experiment of For the experiment of rolling an ordinary die:rolling an ordinary die:

P(even number) =P(even number) =

P(result less than four) P(result less than four)

P(not getting a two) =P(not getting a two) =

3 = 1

6 2

3 = 1

6 2 5

6

Complement of Event AComplement of Event AComplementComplement is not the same as is not the same as ComplimentCompliment

the event the event not Anot A

Probability of a Probability of a ComplementComplement

P(not A) = 1 – P(A)P(not A) = 1 – P(A)

Probability of a Complement

If the probability that it will snow today is 30%,

P(It will not snow) = 1 – P(snow) = 1 – 0.30 = 0.70

Probabilities of an Event and its Probabilities of an Event and its ComplementComplement

Denote the probability of an event by Denote the probability of an event by the letter p. the letter p.

Denote the probability of the Denote the probability of the complement of the event by the complement of the event by the letter q.letter q.

p + q must equal 1p + q must equal 1 q = 1 - pq = 1 - p

Probability Probability Related to StatisticsRelated to Statistics

Probability makes statements about Probability makes statements about what will occur when samples are what will occur when samples are drawn from a drawn from a knownknown population. population.

Statistics describes how samples are Statistics describes how samples are to be obtained and how inferences to be obtained and how inferences are to be made about are to be made about unknownunknown populations.populations.