Download presentation

Presentation is loading. Please wait.

Published byLester Riley Modified over 4 years ago

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

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google