Chapter 6 Lesson 6.1 Probability 6.1: Chance Experiments and Events

Chance experiment – any activity or situation in which there is uncertainty about which of two or more plausible outcomes will result. Suppose two six-sided die is rolled and they both land on sixes. Or a coin is flipped and it lands on heads. Or record the color of the next 20 cars to pass an intersection. These would be examples of chance experiments.

Sample space - the collection of all possible outcomes of a chance experiment Suppose a six-sided die is rolled. The possible outcomes are that the die could land with 1 dot up or 2, 3, 4, 5, or 6 dots up. S = {1, 2, 3, 4, 5, 6} This would be an example of a sample space. “ S ” stands for sample space. We use set notation to list the outcomes of the sample space. The sum of the probabilities of the outcomes in the sample space equals ____.

Suppose two coins are flipped. The sample space would be: S = {HH, HT, TH, TT} Where H = heads and T = tails H T H T H T We can also use a Tree Diagram to represent a sample space. HT We follow the branches out to show an outcome.

Event - any collection of outcomes (subset) from the sample space of a chance experiment Suppose a six-sided die is rolled. The outcome that the die would land on an even number would be E = {2, 4, 6} This would be an example of an event. We typically use capital letters to denote an event.

Complement - Consists of all outcomes that are not in the event Suppose a six-sided die is rolled. The event that the die would land on an even number would be E = {2, 4, 6} What would the event be that is the die NOT landing on an even number? E C = {1, 3, 5} This is an example of complementary events. The superscript “ C ” stands for complement E ’ and E also denote the complement of E The sum of the probabilities of complementary events equals ______.

These complementary events can be shown on a Venn Diagram. E = {2, 4, 6} and E C = {1, 3, 5} Let the rectangle represent the sample space. Let the circle represent event E. Let the shaded area represent event not E.

Suppose a six-sided die is rolled. The event that the die would land on an even number would be E = {2, 4, 6} The event that the die would land on a prime number would be P = {2, 3, 5} What would be the event E or P happening? E or P = {2, 3, 4, 5, 6} This is an example of the union of two events.

The union of A or B - consists of all outcomes that are in at least one of the two events, that is, in A or in B or in both. This symbol means “ union ” Consider a marriage or union of two people – when two people marry, what do they do with their possessions ? The bride takes all her stuff & the groom takes all his stuff & they pool it together! And live happily ever after! This is similar to the union of A and B. All of A and all of B are put together!

Let ’ s revisit rolling a die and getting an even or a prime number... E or P = {2, 3, 4, 5, 6} Another way to represent this is with a Venn Diagram. Even number Prime number E or P would be any number in either circle. Why is the number 1 outside the circles?

Suppose a six-sided die is rolled. The event that the die would land on an even number would be E = {2, 4, 6} The event that the die would land on a prime number would be P = {2, 3, 5} What would be the event E and P happening? E and P = {2} This is an example of the intersection of two events.

The intersection of A and B - consists of all outcomes that are in both of the events This symbol means “ intersection ”

Let ’ s revisit rolling a die and getting an even or a prime number... E and P = {2} To represent this with a Venn Diagram: E and P would be ONLY the middle part that the circles have in common

Mutually exclusive (or disjoint) events - two events have no outcomes in common; two events that NEVER happen simultaneously Suppose a six-sided die is rolled. Consider the following 2 events: A = {2} B = {6} On a single die roll, is it possible for A and B to happen at the same time? These events are mutually exclusive.

A Venn Diagram for the roll of a six-sided die and the following two events: A = {2} B = {6} A and B are mutually exclusive (disjoint) since they have no outcomes in common The intersection of A and B is empty!

Practice with Venn Diagrams On the following four slides you will find Venn Diagrams representing the students at your school. Some students are enrolled in Statistics, some in Calculus, and some in Band. For the next four slides, indicate what relationships the shaded regions represent.

Calculus or Band StatisticsCalculus Band

Statistics or Band and not Calculus StatisticsCalculus Band

Com Sci Statistics and Band and not Calculus StatisticsCalculus Band

Statistics and not (Band or Calculus) StatisticsCalculus Band

M&M Activity

Homework Pg.326: #6.3, 6.5, 6.8, 6.9, 6.11, 6.12 Reading Notes 6.2