Section 8 1 Probability Models and Rules Special

Section 8. 1 - Probability Models and Rules Special Topics

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.

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/gilbert 139343/java 04_s. html

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.

Examples �Let’s say we roll a die and flip a coin. Create the sample space to show all possible outcomes. �S = {H 1, H 2, H 3, H 4, H 5, H 6, T 1, T 2, T 3, T 4, T 5, T 6}. There are 12 outcomes (2 x 6). �A sample space that is unusually long can be truncated: S = {H 1, H 2, …T 5, T 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.

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.

Still More Definitions �Complement of an Event: The complement of an event A is the event that A does not occur, written as AC. �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!

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(AC) = 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.

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 + Ac. Recall that your sample space has a probability of 1, so A + Ac = 1.

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.

Homework �Worksheet 8. 1