I. Randomness (IPS section 4.1 pages 282-287)

A. Random – We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions. Random does NOT mean haphazard. It is a description of a kind of order that emerges only in the long run.

B. Probability – The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of independent repetitions. That is, probability is long-term relative frequency.

II. Probability Models (IPS section 4.2 pages 287-305)

A. Sample Space – The sample space S of a random phenomenon is the set of all possible outcomes.

B. Event – An event is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space.

C. Probability Rules:

1. Any probability is a number between 0 and 1. 0 P(A) 1

2. All possible outcomes together must have probability 1. If S is the sample space in a probability model, then P(S) = 1.

3. The probability that an event does not occur is 1 minus the probability that the event does occur. The complement of any event A is the event that A does not occur, written as Ac. The complement rule states that P(Ac) = 1 – P(A)

4. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. Two events A and B are disjoint if they have no outcomes in common and so can never occur simultaneously. If A and B are disjoint,

P(A or B) = P(A) + P(B)

This is the addition rule for disjoint events. Disjoint events are also called mutually exclusive events.

5. Independence and the Multiplication Rule – Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent, then

P(A and B) = P(A) * P(B)

This is the multiplication rule for independent events. If A and B are NOT independent, then P(A and B) P(A)P(B).

NOTE: disjoint events can NOT be independent since the probability of them both happening is zero, P(A and B) = 0 for disjoint events (p. 296).

D. Venn Diagram – A picture that shows the sample space S as a rectangular area and events as areas within S.

Figure 4.2 Venn diagram showing disjoint events A and B Figure 4.3 Venn diagram showing the complement Ac of an event A

Figure 4.4 Venn diagram showing the event {A and B}. This event consists of outcomes common to A and B

E. Probabilities in a Finite Sample Space – Assign a probability to each individual outcome. These probabilities must be numbers between 0 and 1 must have sum 1. The probability of any event is the sum of the probabilities of the outcome making up the event.

F. Equally Likely Outcomes – If a random phenomenon has k possible outcomes, all equally likely, then each individual outcome has probability 1/k. The probability of any event A is :

P(A) = count of outcomes in A count of outcomes in A count of outcomes in S k

Note: a random variable (see below) that has equally likely outcomes has a uniform distribution.

III. Random Variables (IPS section 4.3 pages 305 – 318)

A. Random Variable – a random variable is a variable whose value is a numerical.

B. Discrete Random Variable – A discrete random variable has a finite number of possible values. The probability distribution of X lists the values and their probabilities:

Value of X x1 x2 x3 … xk Probability p1 p2 p3 … pk The probabilities pi must satisfy two requirements:

1. Every probability pi is a number between 0 and 1.

2. p1 + p2 +…+ pk = 1.

Find the probability of any event by adding the probabilities pi of the particular values xi that make up the event (as in IIE above).

C. Continuous Random Variable – A continuous random variable X takes all values in an