Chapter 14 From Randomness to Probability

Dealing with Random Phenomena A random phenomenon: if we know what outcomes could happen, but not which particular values will happen. A single attempt of a random phenomenon is called a trial. At each trial, the value measured, observed, or reported is called an outcome. The combination of outcomes is an event. The collection of all possible outcomes for an event is called the sample space.

Example/Notation Experiment: rolling a dice once Sample Space S = {1, 2, 3, 4, 5, 6} Possible Events: A = outcome < 4 = {1, 2, 3} B = outcome is even = {2, 4, 6} C = score is 7 = null space A U B = outcome is < 4 or even or both = {1, 2, 3, 4, 6} A∩ B = outcome is < 4 and even = {2} A C = event A does NOT occur = {4, 5, 6}

The Law of Large Numbers Independence: when one event occurring does not change/influence the probability that the other event occurs. - For example, coin flips are independent. Relative Frequency: # of times an event occurs divided by total # of trials Law of Large Numbers (or Probability of an event): the long-run relative frequency (if an experiment is repeated many, many times, the outcome value converges to the true probability.

The probability of an event is the number of outcomes in the event divided by the total number of possible outcomes. P(A) = # of outcomes in A # of possible outcomes

Note: Don’t count on Law of Averages (that an outcome of an event that hasn’t occurred in many trials is “due” to occur). Also, not concerned with personal (subjective) probability: your own personal belief of the probability.

Formal Probability 1.Two requirements for a probability: – A probability is a number between 0 and 1. – For any event A, 0 ≤ P(A) ≤ 1.

2.Probability Assignment Rule: – The probability of the set of all possible outcomes of a trial must be 1. – P(S) = 1 (S represents the set of all possible outcomes.)

3.Complement Rule:  The set of outcomes that are not in the event A is called the complement of A, denoted A C.  The probability of an event occurring is 1 minus the probability that it doesn’t occur: P(A) = 1 – P(A C )  Sometimes it’s easier to use the complement of an event we’re looking for.

4.Addition Rule: Events that have no outcomes in common (and cannot occur at the same time) are called disjoint (or mutually exclusive). P(A  B) = 0

4.Addition Rule (cont.): – For two disjoint events A and B, the probability that one or the other occurs is: P(A  B) = P(A) + P(B)

4. Addition Rule : – Always Check for the Disjoint Assumption before using the Addition Rule.

5.Multiplication Rule: – For two independent events A and B, the probability that both A and B occur is: P(A  B) = P(A)  P(B)

5.Multiplication Rule (cont.): ** Two independent events A and B are not disjoint. ** And disjoint events cannot be independent. Ex: Choosing a student who’s a Junior and Senior? Zero probability – so disjoint. Therefore, picking a Junior changes the outcome for picking a Senior to zero – not independent.

5.Multiplication Rule: – Always Check for the Independence Assumption before using this Multiplication Rule. – If A and B are not independence, there will be an adjustment made to the formula in ch.15.

Notation alert: You might see the following: – P(A or B) instead of P(A  B) – P(A and B) instead of P(A  B)

What Can Go Wrong? Beware of probabilities that don’t add up to 1. – To be a legitimate probability distribution, the sum of the probabilities for all possible outcomes must total 1. Don’t add probabilities of events if they’re not disjoint. – Events must be disjoint to use the Addition Rule.

Don’t multiply probabilities of events if they’re not independent. – The multiplication of probabilities of events that are not independent is one of the most common errors people make in dealing with probabilities. Don’t confuse disjoint and independent—disjoint events can’t be independent.