Chapter 6: Probability

Section 6.1: Chance Experiments and Events

Chance Experiment – An activity or situation in which there is uncertainty about which of two or more possible outcomes will result. (example: Flipping a coin, picking an ace in a deck of cards so you can be mafia, rolling dice)

Sample Space – The collection of all possible outcomes of a chance experiment

Example Consider a chance experiment designed to see whether men and women have different shopping preferences when buying a CD. Four types of music are sold: classical, rock, country and other. The possible outcomes are listed on the next slide:

A male buying classical A female buying classical A male buying rock A female buying rock A male buying country A female buying country A male buying other A female buying other

Sample Space: {(male, classical), (female, classical), (male, rock), (female, rock), (male, country), (female, country), (male, other), (female, other)}

Tree Diagram

Another type of tree diagram

Event – Any collection of outcomes from the sample space of a chance experiment Simple Event – An event consisting of exactly one outcome

Example Reconsider the example before with the CD Let M = male, F = female, C = classical, R = Rock, K = country, O = other Sample Space: {MC, FC, MR, FR, MK, FK, MO, FO}

We have eight outcomes, eight simple events E1 = MC, E2 = FC, E3 = MR, E4 = FR E5 = MK, E6 = FK, E7 = MO, E8 = FO

One event of interest could be classical music. A symbolic description: Classical = {MC, FC} Another event is that the buyer is male. A symbolic description: male = {MC, MR, MK, MO}

Let A and B denote two events 1. The event not A consists of all experimental outcomes that are not in event A. Not A is sometimes called the complement of A is usually denoted by:

2. The event A or B consists of all experimental outcomes that are in at least one of the two events, that is, in A or B or in both of these. A or B is called the union of the two events and is also denoted by:

3. The event A and B consists of all experimental outcomes that are in both the event A and the event B. A and B is called the intersection of the two events and is also denoted by:

Disjoint or mutually exclusive – Two events that have no common outcome Venn Diagram – An informal picture that is used to identify relationships

Venn Diagrams

Example: Turning Directions An observer stands at the bottom of a freeway off-ramp and records the turning direction (L = left of R = Right) of each of three successive vehicles. The sample space contains eight outcomes: {LLL, RLL, LRL, LLR, RRL, RLR, LRR,RRR}

Each of these outcomes determines a simple event. Other events include A = event that exactly one of the cars turns right {RLL, LRL, LLR} B = event that at most one of the cars turns right {LLL, RLL, LRL, LLR} C = even that all cars turn in the same direction {LLL, RRR}

Example - continued not C = Cc = even that not all cars turn in the same direction ={RLL, LRL, LLR, RRL, RLR, LRR} A or C = A ∪ C = event that exactly one of the cars turns right or all cars turn in the same direction = {RLL, LRL, LLR, LLL, RRR} B and C = B ∩ C = event that at most one car turns right and all cars turn in the same direction = {LLL}

Let A1, A2, …, Ak denote k events The event A1 or A2 or … or Ak consists of all outcomes in at least one of the individual events A1, A2, …Ak The event A1 and A2 and … and Ak consists of all outcomes that are simultaneously in every one of the individual events A1, A2, …, Ak These k events are disjoint if no two of them have any common outcomes

Venn Diagram

Example Consider the following events: A = {(1,1) (2,2) (3,3) (4,4)} B = {(1,1) (1,2) (1,3) (1,4) (2,1) (3,1) (4,1)} C = {(3,1) (2,2) (1,3)} D = {(3,3) (3,4) (4,3)} E = {(1,1) (1,3) (2,2) (3,1) (4,2) (3,3) (2,4) (4,4)} F = {(1,1) (1,2) (2,1)}

Then A or C or D = {(1,1) (2,2) (3,3) (4,4) (3,1) (1,3) (3,4) (4,3)} The outcome (3,1) is contained in each of the events B, C, and E as is the outcome (1,3). These are the only two common outcomes, so B and C and E = {(3,1) (1,3)} The events C, D, and F are disjoint because no outcome in any one of these events is contained in either of the other two events.