Section 8.1 - Probability Models and Rules

Special Topics

DefinitionsRandom (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 CoinIndividual 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.

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More DefinitionsProbability: 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.

ExamplesLet’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}.

Tree DiagramAnother 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 DefinitionsComplement 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 ProbabilitiesAny 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 DiagramA 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 DiagramThis 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.

HomeworkWorksheet 8.1