Probability Probability is a numerical measurement of likelihood of an event. Probability is a numerical measurement of likelihood of an event. The probability of any event is a number between zero and one. The probability of any event is a number between zero and one. Events with probability close to one are more likely to occur. Events with probability close to one are more likely to occur. If an event has probability equal to one, the event is certain to occur. If an event has probability equal to one, the event is certain to occur.

Probability Notation If A represents an event, P(A) P(A) represents the probability of A. represents the probability of A.

Three methods to find probabilities: Intuition Intuition Relative frequency Relative frequency Equally likely outcomes Equally likely outcomes

Intuition method based upon our level of confidence in the result Example: I am 95% sure that I will attend the party.

Probability as Relative Frequency Probability of an event = the fraction of the time that the event occurred in the past = the fraction of the time that the event occurred in the past = f f n where f = frequency of an event n = 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:

Equally likely outcomes No one result is expected to occur more frequently than any other.

Probability of an event when outcomes are equally likely =

Example of Equally Likely Outcome Method When rolling a die, the probability of getting a number less than three =

Law of Large Numbers In the long run, as the sample size increases and increases, the relative frequencies of outcomes get closer and closer to the theoretical (or actual) probability value.

Example of Law of Large Numbers Flip a coin 10 times and record the outcomes. (Do it now…ok 1 person do it…no need for everyone to throw money around!) Flip a coin 10 times and record the outcomes. (Do it now…ok 1 person do it…no need for everyone to throw money around!) Flip a coin 20 times and record the outcomes. (Yes, do this now also!) Flip a coin 20 times and record the outcomes. (Yes, do this now also!) Flip a coin 30 times and record the outcomes. (Don’t skip this step!) Flip a coin 30 times and record the outcomes. (Don’t skip this step!)

Example of Law of Large Numbers What happens to the P(heads) vs P(tails) as the sample space increases? What happens to the P(heads) vs P(tails) as the sample space increases? How does the actual result compare to the theoretical result? What did happen vs what should have happened? How does the actual result compare to the theoretical result? What did happen vs what should have happened?

Statistical Experiment activity that results in a definite outcome

Sample Space set of all possible outcomes of an experiment

Sample Space for the rolling of an ordinary die: 1, 2, 3, 4, 5, 6

For the experiment of 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 = =

Complement of Event A Complement is not the same as Compliment the event not A

Probability of a Complement 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 Complement Denote the probability of an event by the letter p. Denote the probability of an event by the letter p. Denote the probability of the complement of the event by the letter q. Denote the probability of the complement of the event by the letter q. p + q must equal 1 p + q must equal 1 q = 1 - p q = 1 - p

Probability Related to Statistics Probability makes statements about what will occur when samples are drawn from a known population. Probability makes statements about what will occur when samples are drawn from a known population. Statistics describes how samples are to be obtained and how inferences are to be made about unknown populations. Statistics describes how samples are to be obtained and how inferences are to be made about unknown populations.