1. Special Topics

2. Definitions Random (not haphazard): A phenomenon or trial is said to be random if individual outcomes are uncertain but the long-term pattern of many individual outcomes is predictable. Randomness is a kind of order, an order that emerges only in the long run, over many repetitions. Examples: hair color, the spread of epidemics, outcomes of games of chance, flipping coins, etc.

3. Example: Tossing a Coin Individual coin tosses are not predictable, so it would not be impossible to flip coins and see 5 consecutive “heads”. However, if we are able to flip a coin indefinitely, we would see the true proportion of heads emerge, which is p =.5. This is a “long-run” random probability. http://www.wiley.com/college/mat/gilbert139343/java/ java04_s.html http://www.wiley.com/college/mat/gilbert139343/java/ java04_s.html

4. More Definitions 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 repetitions. Sample Space: The sample space, or “S” of a random phenomenon is the set of all possible outcomes that cannot be broken down further into simpler components. Event: An event is any outcome or any set of outcomes of a random phenomenon. That is, an event is a subset of the sample space.

5. Examples Let’s say we roll a die and flip a coin. Create the sample space to show all possible outcomes. S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}. There are 12 outcomes (2 x 6). A sample space that is unusually long can be truncated: S = {H1, H2,…T5, T6}.

6. Tree Diagram Another way to determine a sample space is with a tree diagram: Thus, the sample space is S = {HH, HT, TH, TT}. Note that there are 4 outcomes, but only 3 events. This would be done on paper with symbols, not coins.

7. Definitions Continued… Probability Model: A probability model is a mathematical description of a random phenomenon consisting of two parts: a sample space S and a way of assigning probabilities to events. There are two ways to arrive at probabilities: Empirical Probabilities: These are probabilities arrived at through repeating an experiment, such as flipping a coin many times and recording the proportion of heads observed. Theoretical Probabilities: These are probabilities arrived at through formulas and calculations.

8. Still More Definitions Complement of an Event: The complement of an event A is the event that A does not occur, written as A C. Disjoint Events: Two events are disjoint events if they have no outcomes in common (they can’t happen at the same time). Disjoint events are also called mutually exclusive events. Independent Events: Two events are independent events if the occurrence of one event has no effect on the probability of the occurrence of the other event. *Note*! Independence and Disjoint (mutually exclusive) don’t mean the same thing!

9. Rules for Probabilities Any probability is a number between 0 and 1 inclusive. So… 0 ≤ P(E)≤ 1. P(E) means “probability of an event.” All possible outcomes together must have probability of 1. This means that the sum of all the probabilities in a sample space equals 1. The probability that an event does not occur is 1 minus the probability that the event does occur. This is saying that P(A C ) = 1 – P(A). If two events are disjoint, the probability that one or the other occurs is the sum of their individual probabilities. The word “or” in probability means “+” or add.

10. Venn Diagram A Venn Diagram is a visual which helps to visualize probability situations. The following is a Venn Diagram for the Complement Rule. Thus, S = A + A c. Recall that your sample space has a probability of 1, so A + A c = 1.

11. Another Venn Diagram This Venn Diagram illustrates the Rule for Disjoint Events: Thus, the probability of A or B equals P(A) + P(B). We will cover non-disjoint events tomorrow.

12. Homework Worksheet 8.1